Twisted Modules and G-equivariantization in Logarithmic Conformal Field Theory

Abstract

A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories ${\mathcal {C}}$ of modules for the fixed-point vertex operator subalgebra $V^G$ of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every $V^G$-module in ${\mathcal {C}}$ with a unital and associative V-action is a direct sum of g-twisted V-modules for possibly several $g\in G$, that the category of all such twisted V-modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category ${\mathcal {C}}$ of $V^G$-modules. This generalizes results of Kirillov and Müger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether $V^G$ is strongly rational if V is strongly rational. We show that $V^G$ is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and $V^G$ is $C_2$-cofinite.

Appendices

Proof of Theorem 2.10

We use the notation and setting of Sect. 2.2. We have already seen that ${{\,\mathrm{Rep}\,}}^G V$ is an ${\mathbb {F}}$-additive supercategory with a G-grading, and Corollary 2.5 shows that ${{\,\mathrm{Rep}\,}}^G V$ has a monoidal structure compatible with the grading. It remains to show that the G-action and braiding isomorphisms on ${{\,\mathrm{Rep}\,}}^G V$ discussed in Sect. 2.2 are well defined and satisfy the required properties.

First we show that $T_g$ is a superfunctor on ${{\,\mathrm{Rep}\,}}^G V$. To show that $(T_g(W), \mu _{T_g(W)})$ is an object of ${{\,\mathrm{Rep}\,}}V$, we first verify associativity:

$$\begin{aligned} \mu _{T_g(W)}(1_V\boxtimes \mu _{T_g(W)})&= \mu _W(1_V\boxtimes \mu _W)(g^{-1}\boxtimes (g^{-1}\boxtimes 1_W)) \\&=\mu _W(\mu _V\boxtimes 1_W){\mathcal {A}}_{V,V,W}(g^{-1}\boxtimes (g^{-1}\boxtimes 1_W)) \\&=\mu _W(g^{-1}\boxtimes 1_V)(\mu _V\boxtimes 1_W){\mathcal {A}}_{V,V,W} \\&=\mu _{T_g(W)}(\mu _V\boxtimes 1_V){\mathcal {A}}_{V,V,T_g(W)}, \end{aligned}$$

using $g^{-1}\mu _V=\mu _V(g^{-1}\boxtimes g^{-1})$. For the unit property,

$$\begin{aligned} \mu _{T_g(W)}(\iota _V\boxtimes 1_W) l_{T_g(W)}^{-1}= & {} \mu _W(g^{-1}\boxtimes 1_W)(\iota _V\boxtimes 1_W) l_W^{-1} \\= & {} \mu _W(\iota _V\boxtimes 1_W) l_W^{-1}=1_W=1_{T_g(W)} \end{aligned}$$

using $g^{-1}\iota _V=\iota _V$. A morphism $f: W_1\rightarrow W_2$ in ${{\,\mathrm{Rep}\,}}V$ is still a morphism from $T_g(W_1)$ to $T_g(W_2)$ because

$$\begin{aligned} f\mu _{T_g(W_1)}= & {} f\mu _{W_1}(g^{-1}\boxtimes 1_W)=\mu _{W_2}(1_V\boxtimes f)(g^{-1}\boxtimes 1_W) \\= & {} \mu _{W_2}(g^{-1}\boxtimes 1_W)(1_V\boxtimes f) =\mu _{T_g(W_2)}(1_V\boxtimes f). \end{aligned}$$

Because $g^{-1}$ is even, we avoid a sign factor in the third equality here. Clearly $T_g$ induces an even linear map on morphisms, so $T_g$ is a superfunctor on ${{\,\mathrm{Rep}\,}}\,V$. Then $T_g$ restricts to a superfunctor on ${{\,\mathrm{Rep}\,}}^G V$ because if $(W,\mu _W)$ is an h-twisted V-module for $h\in G$, then $(T_g(W),\mu _{T_g(W)})$ is a $g h g^{-1}$-twisted V-module. Indeed,

$$\begin{aligned} \mu _{T_g(W)}(ghg^{-1}\boxtimes 1_{T_g(W)}){\mathcal {M}}_{V,T_g(W)}&=\mu _W(g^{-1}\boxtimes 1_W)(ghg^{-1}\boxtimes 1_W){\mathcal {M}}_{V,W} \\&=\mu _W(h\boxtimes 1_W){\mathcal {M}}_{V,W}(g^{-1}\boxtimes 1_W) \\&=\mu _W(g^{-1}\boxtimes 1_W) =\mu _{T_g(W)}, \end{aligned}$$

using the naturality of the monodromy isomorphisms in the second equality.

Next we construct the even natural isomorphism $\tau _g: T_g\circ \boxtimes _V\rightarrow \boxtimes _V\circ (T_g\times T_g)$. For objects $W_1$, $W_2$ in ${{\,\mathrm{Rep}\,}}V$, recall that $(W_1\boxtimes _V W_2, I_{W_1,W_2})$ is the cokernel of the morphism $\mu ^{(1)}_{W_1,W_2}-\mu ^{(2)}_{W_1,W_2}$ and similarly for $(T_g(W_1)\boxtimes _V T_g(W_2), I_{T_g(W_1),T_g(W_2)})$. We claim that there are unique morphisms

$$\begin{aligned}&\tau _{g; W_1, W_2}: W_1\boxtimes _V W_2\rightarrow T_g(W_1)\boxtimes _V T_g(W_2), \\&{\widetilde{\tau }}_{g; W_1,W_2}: T_g(W_1)\boxtimes _V T_g(W_2)\rightarrow W_1\boxtimes _V W_2 \end{aligned}$$

in ${\mathcal {SC}}$ such that the diagrams

commute. This follows from the universal properties of the cokernels and the equalities

$$\begin{aligned}&I_{T_g(W_1),T_g(W_2)} (\mu _{W_1,W_2}^{(1)}-\mu _{W_1,W_2}^{(2)})\\&\quad = I_{T_g(W_1),T_g(W_2)}(\mu _{W_1,W_2}^{(1)}-\mu _{W_1,W_2}^{(2)})(g^{-1}\boxtimes 1_{W_1\boxtimes W_2})(g\boxtimes 1_{W_1\boxtimes W_2}) \\&\quad =I_{T_g(W_1),T_g(W_2)} (\mu _{T_g(W_1),T_g(W_2)}^{(1)} -\mu _{T_g(W_1),T_g(W_2)}^{(2)})(g\boxtimes 1_{W_1\boxtimes W_2})= 0 \end{aligned}$$

and

$$\begin{aligned}&I_{W_1, W_2}(\mu _{T_g(W_1),T_g(W_2)}^{(1)}-\mu _{T_g(W_1),T_g(W_2)}^{(2)}) \\&\quad = I_{W_1,W_2}(\mu _{W_1,W_2}^{(1)}-\mu _{W_1, W_2}^{(2)})(g^{-1}\boxtimes 1_{W_1\boxtimes W_2}) \\&\quad = 0. \end{aligned}$$

These equalities use the definitions of the $\mu ^{(i)}$, the naturality of associativity and braiding isomorphisms in ${\mathcal {SC}}$, and the evenness of all morphisms involved. Now $\tau _{g; W_1, W_2}$ and ${\widetilde{\tau }}_{g; W_1, W_2}$ are mutual inverses: because

$$\begin{aligned} {\widetilde{\tau }}_{g; W_1, W_2} \tau _{g; W_1, W_2} I_{W_1,W_2} ={\widetilde{\tau }}_{g; W_1,W_2} I_{T_g(W_1),T_g(W_2)} =I_{W_1,W_2} \end{aligned}$$

and $I_{W_1,W_2}$ is surjective, ${\widetilde{\tau }}_{g; W_1, W_2} \tau _{g; W_1, W_2}= 1_{W_1\boxtimes _V W_2}$, and similarly $\tau _{g; W_1, W_2}{\widetilde{\tau }}_{g; W_1,W_2}= 1_{T_g(W_1)\boxtimes _V T_g(W_2)}$. Also, $\tau _{g; W_1, W_2}$ is even because $I_{W_1,W_2}$ and $I_{T_g(W_1),T_g(W_1)}$ are even and surjective.

Now we show that $\tau _{g; W_1, W_2}$ is a morphism in ${{\,\mathrm{Rep}\,}}V$ from $T_g(W_1\boxtimes _V W_2)$ to $T_g(W_1)\boxtimes _V T_g(W_2)$. Then its inverse ${\widetilde{\tau }}_{g; W_1, W_2}: T_g(W_1)\boxtimes _V T_g(W_2)\rightarrow T_g(W_1\boxtimes _V W_2)$ will also be a morphism in ${{\,\mathrm{Rep}\,}}V$. This uses the commutative diagrams

for $i=1$ or $i=2$ and

The top compositions in these two diagrams agree because $I_{T_g(W_1),T_g(W_2)}$ is an intertwining operator and because $\mu ^{(i)}_{W_1,W_2}(g^{-1}\boxtimes 1_{W_1\boxtimes W_2})=\mu ^{(i)}_{T_g(W_1),T_g(W_2)}$ for $i=1,2$. Thus

$$\begin{aligned} \tau _{g; W_1,W_2}\mu _{T_g(W_1\boxtimes _V W_2)}(1_V\boxtimes I_{W_1,W_2}) =\mu _{T_g(W_1)\boxtimes _V T_g(W_2)}(1_V\boxtimes \tau _{g; W_1,W_2})(1_V\boxtimes I_{W_1,W_2}) \end{aligned}$$

as well. Since $I_{W_1, W_2}$ is a surjective cokernel morphism and $V\boxtimes \bullet$ is right exact, $1_V\boxtimes I_{W_1,W_2}$ is surjective as well and it follows that $\tau _{g; W_1,W_2}$ is a morphism in ${{\,\mathrm{Rep}\,}}V$.

Next we show that the $\tau _{g; W_1, W_2}$ define a natural isomorphism, that is, for morphisms $f_1: W_1\rightarrow {\widetilde{W}}_1$ and $f_2: W_2\rightarrow {\widetilde{W}}_2$ in ${{\,\mathrm{Rep}\,}}V$,

$$\begin{aligned} \tau _{g; {\widetilde{W}}_1,{\widetilde{W}}_2} T_g(f_1\boxtimes _V f_2)=(T_g(f_1)\boxtimes _V T_g(f_2)) \tau _{g; W_1, W_2}. \end{aligned}$$

This follows from the commutative diagrams

and

as well as the surjectivity of $I_{W_1,W_2}$.

The even natural isomorphism $\tau _g$ needs to be compatible with the associativity isomorphisms in the sense that the diagram

commutes for any objects $W_1$, $W_2$, and $W_3$ in ${{\,\mathrm{Rep}\,}}V$. For the proof, recall that $T_g(W)=W$ as objects of ${\mathcal {SC}}$ and $T_g(f)=f$ when $(W,\mu _W)$ is an object and f is a morphism in ${{\,\mathrm{Rep}\,}}V$. Consider the composition

$$\begin{aligned}&W_1\boxtimes (W_2\boxtimes W_3) \xrightarrow {1_{W_1}\boxtimes I_{W_2,W_3}} W_1\boxtimes (W_2\boxtimes _V W_3)\xrightarrow {I_{W_1,W_2\boxtimes _V W_3}} W_1\boxtimes _V(W_2\boxtimes _V W_3) \\&\quad \xrightarrow {{\mathcal {A}}^V_{W_1,W_2,W_3}} (W_1\boxtimes _V W_2)\boxtimes _V W_3\xrightarrow {\tau _{g; W_1\boxtimes _V W_2, W_3}} T_g(W_1\boxtimes _V W_2)\boxtimes _V T_g(W_3) \\&\quad \xrightarrow {\tau _{g; W_1, W_2}\boxtimes _V 1_{T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3). \end{aligned}$$

By the definition of the associativity isomorphisms in ${{\,\mathrm{Rep}\,}}V$, this equals

$$\begin{aligned} W_1\boxtimes (W_2\boxtimes W_3)&\xrightarrow {{\mathcal {A}}_{W_1,W_2,W_3}} (W_1\boxtimes W_2)\boxtimes W_3\xrightarrow {I_{W_1,W_2}\boxtimes 1_{W_3}} (W_1\boxtimes _V W_2)\boxtimes W_3 \\&\xrightarrow {I_{W_1\boxtimes _V W_2, W_3}} (W_1\boxtimes _V W_2)\boxtimes _V W_3 \\&\xrightarrow {\tau _{g; W_1\boxtimes _V W_2, W_3}} T_g(W_1\boxtimes _V W_2)\boxtimes _V T_g(W_3) \\&\xrightarrow {\tau _{g; W_1, W_2}\boxtimes _V 1_{T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3). \end{aligned}$$

Then the definition of $\tau _{g; W_1\boxtimes _V W_2, W_3}$ implies that we get

$$\begin{aligned} W_1\boxtimes (W_2\boxtimes W_3)&\xrightarrow {{\mathcal {A}}_{W_1,W_2,W_3}} (W_1\boxtimes W_2)\boxtimes W_3\xrightarrow {I_{W_1,W_2}\boxtimes 1_{W_3}} (W_1\boxtimes _V W_2)\boxtimes W_3\nonumber \\&\xrightarrow {I_{T_g(W_1\boxtimes _V W_2),T_g(W_3)}} T_g(W_1\boxtimes _V W_2)\boxtimes _V T_g(W_3) \nonumber \\&\xrightarrow {\tau _{g; W_1, W_2}\boxtimes _V 1_{T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3). \end{aligned}$$

(A.1)

From the definition of the tensor product of morphisms in ${{\,\mathrm{Rep}\,}}V$,

$$\begin{aligned} (\tau _{g; W_1, W_2}\boxtimes _V 1_{T_g(W_3)}) I_{T_g(W_1\boxtimes _V W_2),T_g(W_3)}=I_{T_g(W_1)\boxtimes _V T_g(W_2),T_g(W_3)}(\tau _{g; W_1,W_2}\boxtimes 1_{T_g(W_3)}), \end{aligned}$$

and then the definition of $\tau _{g; W_1, W_2}$ implies that (A.1) becomes

$$\begin{aligned} W_1\boxtimes (W_2\boxtimes W_3)&\xrightarrow {{\mathcal {A}}_{W_1,W_2,W_3}} (W_1\boxtimes W_2)\boxtimes W_3 \\&\xrightarrow {I_{T_g(W_1),T_g(W_2)}\boxtimes 1_{W_3}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes T_g(W_3) \\&\xrightarrow {I_{T_g(W_1)\boxtimes _V T_g(W_2), T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3). \end{aligned}$$

Next, the definition of the associativity isomorphisms in ${{\,\mathrm{Rep}\,}}V$ implies that this composition equals

$$\begin{aligned} W_1\boxtimes (W_2\boxtimes W_3)&\xrightarrow {1_{W_1}\boxtimes I_{T_g(W_2),T_g(W_3)}} T_g(W_1)\boxtimes (T_g(W_2)\boxtimes _V T_g(W_3)) \\&\quad \xrightarrow {I_{T_g(W_1),T_g(W_2)\boxtimes _V T_g(W_3)}} T_g(W_1)\boxtimes _V(T_g(W_2)\boxtimes _V T_g(W_3)) \\&\quad \xrightarrow {{\mathcal {A}}^V_{T_g(W_1),T_g(W_2),T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3). \end{aligned}$$

We replace $I_{T_g(W_2),T_g(W_3)}$ with $\tau _{g; W_2,W_3} I_{W_2,W_3}$ and use the definition of tensor product of morphisms in ${{\,\mathrm{Rep}\,}}V$:

$$\begin{aligned} W_1\boxtimes (W_2\boxtimes W_3)&\xrightarrow {1_{W_1}\boxtimes I_{W_2,W_3}} W_1\boxtimes (W_2\boxtimes _V W_3) \\&\xrightarrow {I_{T_g(W_1),T_g(W_2\boxtimes _V W_3)}} T_g(W_1)\boxtimes _V T_g(W_2\boxtimes _V W_3) \\&\xrightarrow {1_{T_g(W_1)}\boxtimes _V \tau _{g; W_2,W_3}} T_g(W_1)\boxtimes _V(T_g(W_2)\boxtimes _V T_g(W_3)) \\&\xrightarrow {{\mathcal {A}}^V_{T_g(W_1),T_g(W_2),T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3). \end{aligned}$$

Finally we use the definition of $\tau _{g; W_1, W_2\boxtimes _V W_3}$ to obtain

$$\begin{aligned} W_1\boxtimes (W_2\boxtimes W_3)&\xrightarrow {1_{W_1}\boxtimes I_{W_2,W_3}} W_1\boxtimes (W_2\boxtimes _V W_3) \\&\xrightarrow {I_{W_1,W_2\boxtimes _V W_3}} W_1\boxtimes _V(W_2\boxtimes _V W_3) \\&\xrightarrow {\tau _{g; W_1, W_2\boxtimes _V W_3}} T_g(W_1)\boxtimes _V T_g(W_2\boxtimes _V W_3) \\&\xrightarrow {1_{T_g(W_1)}\boxtimes _V \tau _{g; W_2,W_3}}T_g(W_1)\boxtimes _V(T_g(W_2)\boxtimes _V T_g(W_3)) \\&\xrightarrow {{\mathcal {A}}^V_{T_g(W_1),T_g(W_2),T_g(W_3)}} (T_g(W_1)\boxtimes _V T_g(W_2))\boxtimes _V T_g(W_3), \end{aligned}$$

and compatibility follows from the surjectivity of $1_{W_1}\boxtimes I_{W_1,W_2}$ and $I_{W_1,W_2\boxtimes _V W_3}$, and hence of their composition.

Now the even morphism $\varphi _g=g: T_g(V)\rightarrow V$ needs to be an isomorphism in ${{\,\mathrm{Rep}\,}}V$. In fact,

$$\begin{aligned} g\mu _{T_g(V)}=g\mu _V(g^{-1}\boxtimes 1_V) =\mu _V(1\boxtimes g) \end{aligned}$$

because g is an automorphism of V. The isomorphism $\varphi _g$ also needs to be compatible with $\tau _g$ and the unit isomorphisms in ${{\,\mathrm{Rep}\,}}V$ in the sense that

$$\begin{aligned} l^V_{T_g(W)}(\varphi _g\boxtimes _V 1_{T_g(W)}) \tau _{g; V, W} = T_g(l^V_W): T_g(V\boxtimes _V W)\rightarrow T_g(W) \end{aligned}$$

(A.2)

and

$$\begin{aligned} r^V_{T_g(W)}(1_{T_g(W)}\boxtimes _V \varphi _g) \tau _{g; W, V} =T_g(r^V_W): T_g(W\boxtimes _V V)\rightarrow T_g(W) \end{aligned}$$

(A.3)

for any object W in ${{\,\mathrm{Rep}\,}}V$. Since $I_{V,W}$ and $I_{W, V}$ are surjective, it is sufficient to show that the equalities in (A.2) and (A.3) hold when both sides are precomposed with

$$\begin{aligned} I_{V,W}: V\boxtimes W\rightarrow V\boxtimes _V W=T_g(V\boxtimes _V W) \end{aligned}$$

and

$$\begin{aligned} I_{W,V}: W\boxtimes V\rightarrow W\boxtimes _V V=T_g(W\boxtimes _V W), \end{aligned}$$

respectively.

For (A.2), we get the composition

$$\begin{aligned} V\boxtimes W&\xrightarrow {I_{V,W}}T_g(V\boxtimes _V W)\xrightarrow {\tau _{g; V, W}} T_g(V)\boxtimes _V T_g(W) \\&\xrightarrow {g\boxtimes _V 1_{T_g(W)}} V\boxtimes _V T_g(W)\xrightarrow {l^V_{T_g(W)}} T_g(W). \end{aligned}$$

Using $\tau _{g; V,W} I_{V,W}=I_{T_g(V),T_g(W)}$ and the definition of the tensor product of morphisms in ${{\,\mathrm{Rep}\,}}V$, this becomes

$$\begin{aligned} V\boxtimes W\xrightarrow {g\boxtimes 1_W} V\boxtimes W\xrightarrow {I_{V,T_g(W)}} V\boxtimes _V T_g(W)\xrightarrow {l^V_{T_g(W)}} T_g(W). \end{aligned}$$

By the definition of the left unit isomorphism in ${{\,\mathrm{Rep}\,}}V$, the last two arrows above can be replaced with $\mu _{T_g(W)}=\mu _W(g^{-1}\boxtimes 1_W)$, so that in total the composition is simply $\mu _W$. But this is

$$\begin{aligned} l^V_W I_{V,W}=T_g(l^V_W) I_{V,W}, \end{aligned}$$

as required. Now for (A.3), we have the composition

$$\begin{aligned} \begin{aligned} W\boxtimes V&\xrightarrow {I_{W,V}} W\boxtimes _V V\xrightarrow {\tau _{g; W, V}} T_g(W)\boxtimes _V T_g(V) \\&\xrightarrow {1_{T_g(W)}\boxtimes _V g} T_g(W)\boxtimes _V V\xrightarrow {r^V_{T_g(W)}} T_g(W). \end{aligned} \end{aligned}$$

Similar to before, this composition is

$$\begin{aligned} W\boxtimes V\xrightarrow {1_W\boxtimes g} W\boxtimes V\xrightarrow {I_{T_g(W),V}} T_g(W)\boxtimes _V V\xrightarrow {r^V_{T_g(W)}} T_g(W). \end{aligned}$$

By definition of $r^V_{T_g(W)}$, this equals

$$\begin{aligned} W\boxtimes V\xrightarrow {1_W\boxtimes g} W\boxtimes V\xrightarrow {{\mathcal {R}}_{V,W}^{-1}} V\boxtimes W\xrightarrow {\mu _{T_g(W)}} T_g(W). \end{aligned}$$

Since $\mu _{T_g(W)}=\mu _W(g^{-1}\boxtimes 1_W)$, naturality of the braiding isomorphisms in ${\mathcal {SC}}$ implies that we get

$$\begin{aligned} W\boxtimes V\xrightarrow {{\mathcal {R}}_{V,W}^{-1}} V\boxtimes W\xrightarrow {\mu _W} W=T_g(W). \end{aligned}$$

By definition, this is $r^V_{W} I_{W,V}=T_g(r^V_W) I_{W,V}$, as desired. This completes the proof that $(T_g, \tau _g, \varphi _g)$ is a tensor endofunctor of ${{\,\mathrm{Rep}\,}}V$, restricting to a tensor endofunctor on ${{\,\mathrm{Rep}\,}}^G V$.

To finish the construction of the G-action on ${{\,\mathrm{Rep}\,}}^G V$, we need to prove that $g\mapsto (T_g, \tau _g, \varphi _g)$ is a group homomorphism. Note first that $(T_1, \tau _1, \varphi _1)$ is the identity functor on ${{\,\mathrm{Rep}\,}}V$ and ${{\,\mathrm{Rep}\,}}^G V$, and we also need to show that $(T_{gh}, \tau _{gh}, \varphi _{gh})$ is the composition of $(T_g, \tau _g, \varphi _g)$ and $(T_h, \tau _h, \varphi _h)$ for $g,h\in G$, that is:

• $T_g(T_h(W,\mu _W))=T_{gh}(W,\mu _W)$ for any object $(W,\mu _W)$ in ${{\,\mathrm{Rep}\,}}V$, and $T_g(T_h(f))=T_{gh}(f)$ for any morphism in ${{\,\mathrm{Rep}\,}}V$.

• $\tau _{g; T_h(W_1),T_h(W_2)} T_g(\tau _{h; W_1,W_2})=\tau _{gh; W_1,W_2}$ for all objects $W_1$ and $W_2$ in ${{\,\mathrm{Rep}\,}}V$.

• $\varphi _g T_g(\varphi _h)=\varphi _{gh}$.

The first point is easy because

$$\begin{aligned} \mu _{T_g(T_h(W))}= & {} \mu _{T_h(W)}(g^{-1}\boxtimes 1_W)=\mu _{W}(h^{-1}\boxtimes 1_W)(g^{-1}\boxtimes 1_W) \\= & {} \mu _W((gh)^{-1}\boxtimes 1_W)=\mu _{T_{gh}(W)} \end{aligned}$$

and because $T_g(T_h(f))=f=T_{gh}(f)$. Also, $\varphi _g T_g(\varphi _h)=gh=\varphi _{gh}$. Then because $\tau _{gh; W_1, W_2}$ is the unique morphism such that

commutes, the commutative diagram

shows that the second point holds as well.

Having constructed the G-action on ${{\,\mathrm{Rep}\,}}^G V$, we now construct the braiding isomorphisms. For objects $W_1$, $W_2$ in ${{\,\mathrm{Rep}\,}}^G V$ with $W_1$ a g-twisted V-module for some $g\in G$, we will show that there are unique morphisms ${\mathcal {R}}^V_{W_1,W_2}$ and $({\mathcal {R}}^V_{W_1,W_2})^{-1}$ such that

commute. Such morphisms would be mutual inverses by the surjectivity of $I_{W_1,W_2}$ and $I_{T_g(W_2),W_1}$, so it remains to show their existence as morphisms in ${\mathcal {SC}}$ and that ${\mathcal {R}}^V_{W_1,W_2}$ is a morphism in ${{\,\mathrm{Rep}\,}}V$.

The existence and uniqueness of the morphisms ${\mathcal {R}}^V_{W_1,W_2}$ and $({\mathcal {R}}^V_{W_1,W_2})^{-1}$ in ${\mathcal {SC}}$ will follow from the universal properties of the cokernels $(W_1\boxtimes _V W_2, I_{W_1,W_2})$ and $(T_g(W_2)\boxtimes _V W_1, I_{T_g(W_2), W_1})$ provided we can show:

$$\begin{aligned}&I_{T_g(W_2), W_1}{\mathcal {R}}_{W_1,W_2}\mu ^{(1)}_{W_1,W_2}=I_{T_g(W_2),W_1} {\mathcal {R}}_{W_1,W_2}\mu ^{(2)}_{W_1,W_2} \end{aligned}$$

(A.4)

$$\begin{aligned}&I_{W_1,W_2}{\mathcal {R}}_{W_1,W_2}^{-1}\mu ^{(1)}_{T_g(W_2), W_1}=I_{W_1,W_2} {\mathcal {R}}_{W_1,W_2}^{-1}\mu ^{(2)}_{T_g(W_2), W_1} \end{aligned}$$

(A.5)

To verify (A.4), we start with $I_{T_g(W_2), W_1}{\mathcal {R}}_{W_1,W_2}\mu ^{(2)}_{W_1,W_2}$, which is the composition

$$\begin{aligned} V\boxtimes (W_1\boxtimes W_2)&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2\xrightarrow {{\mathcal {R}}_{V,W_1}\boxtimes 1_{W_2}} (W_1\boxtimes V)\boxtimes W_2 \\&\xrightarrow {{\mathcal {A}}_{W_1,V,W_2}^{-1}} W_1\boxtimes (V\boxtimes W_2)\xrightarrow {1_{W_1}\boxtimes \mu _{W_2}} W_1\boxtimes W_2 \\&\xrightarrow {{\mathcal {R}}_{W_1,W_2}} W_2\boxtimes W_1\xrightarrow {I_{T_g(W_2)\boxtimes W_1}} T_g(W_2)\boxtimes _V W_1. \end{aligned}$$

By the naturality of the braiding isomorphisms and the hexagon axiom in ${\mathcal {SC}}$, this equals

$$\begin{aligned} V\boxtimes (W_1\boxtimes W_2)&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2\xrightarrow {{\mathcal {M}}_{V,W_1}\boxtimes 1_{W_2}} (V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}^{-1}} V\boxtimes (W_1\boxtimes W_2) \\&\xrightarrow {1_{V}\boxtimes {\mathcal {R}}_{W_1,W_2}} V\boxtimes (W_2\boxtimes W_1)\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}} (V\boxtimes W_2)\boxtimes W_1 \\&\xrightarrow {\mu _{W_2}\boxtimes 1_{W_1}} W_2\boxtimes W_1\xrightarrow {I_{T_g(W_2), W_1}}T_g(W_2)\boxtimes _V W_1. \end{aligned}$$

We replace $\mu _{W_2}$ with $\mu _{T_g(W_2)}(g\boxtimes 1_{W_2})$ and then use the intertwining operator property of $I_{T_g(W_2), W_1}$ and naturality of the associativity isomorphisms:

$$\begin{aligned} V\boxtimes (W_1\boxtimes W_2)&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2\xrightarrow {{\mathcal {M}}_{V,W_1}\boxtimes 1_{W_2}} (V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {(g\boxtimes 1_{W_1})\boxtimes 1_{W_2}}(V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}^{-1}} V\boxtimes (W_1\boxtimes W_2)\xrightarrow {1_{V}\boxtimes {\mathcal {R}}_{W_1,W_2}} V\boxtimes (W_2\boxtimes W_1) \\&\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}} (V\boxtimes W_2)\boxtimes W_1 \\&\xrightarrow {{\mathcal {R}}_{V,W_2}\boxtimes 1_{W_1}} (W_2\boxtimes V)\boxtimes W_1\xrightarrow {{\mathcal {A}}_{W_2,V,W_1}^{-1}} W_2\boxtimes (V\boxtimes W_1) \\&\xrightarrow {1_{W_2}\boxtimes \mu _{W_1}} W_2\boxtimes W_1\xrightarrow {I_{T_g(W_2),W_1}} T_g(W_2)\boxtimes _V W_1. \end{aligned}$$

Now we apply the hexagon axiom and naturality of the braiding in ${\mathcal {SC}}$ to reduce this composition to

$$\begin{aligned} V\boxtimes (W_1\boxtimes W_2)&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2\xrightarrow {{\mathcal {M}}_{V,W_1}\boxtimes 1_{W_2}} (V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {(g\boxtimes 1_{W_1})\boxtimes 1_{W_2}}(V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {\mu _{W_1}\boxtimes 1_{W_2}} W_1\boxtimes W_2\xrightarrow {{\mathcal {R}}_{W_1,W_2}} W_2\boxtimes W_1\xrightarrow {I_{T_g(W_2),W_1}} T_g(W_2)\boxtimes _V W_1. \end{aligned}$$

We replace $\mu _{W_1}(g\boxtimes 1_{W_1}){\mathcal {M}}_{V,W_1}$ with $\mu _{W_1}$ since $W_1$ is a g-twisted V-module, and the resulting composition is $I_{T_g(W_2),W_1}{\mathcal {R}}_{W_1,W_2}\mu ^{(1)}_{W_1,W_2}$, as desired.

Now to prove (A.5), we start with $I_{W_1,W_2}{\mathcal {R}}_{W_1,W_2}^{-1}\mu ^{(1)}_{T_g(W_2),W_1}$, which is the composition

$$\begin{aligned} V\boxtimes (W_2\boxtimes W_1)&\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}} (V\boxtimes W_2)\boxtimes W_1\xrightarrow {(g^{-1}\boxtimes 1_{W_2})\boxtimes 1_{W_1}} (V\boxtimes W_2)\boxtimes W_1 \\&\xrightarrow {\mu _{W_2}\boxtimes 1_{W_1}} W_2\boxtimes W_1\xrightarrow {{\mathcal {R}}_{W_1,W_2}^{-1}} W_1\boxtimes W_2\xrightarrow {I_{W_1,W_2}} W_1\boxtimes _V W_2. \end{aligned}$$

Using naturality of the braiding isomorphisms and the hexagon axiom in ${\mathcal {SC}}$, this becomes

$$\begin{aligned} V\boxtimes (W_2\boxtimes W_1)&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{W_1,W_2}^{-1}} V\boxtimes (W_1\boxtimes W_2)\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {{\mathcal {R}}_{W_1,V}^{-1}\boxtimes 1_{W_2}} (W_1\boxtimes V)\boxtimes W_2 \\&\xrightarrow {{\mathcal {A}}^{-1}_{W_1,V,W_2}} W_1\boxtimes (V\boxtimes W_2)\xrightarrow {1_{W_1}\boxtimes (g^{-1}\boxtimes 1_{W_2})} W_1\boxtimes (V\boxtimes W_2) \\&\xrightarrow {1_{W_1}\boxtimes \mu _{W_2}} W_1\boxtimes W_2\xrightarrow {I_{W_1,W_2}} W_1\boxtimes _V W_2. \end{aligned}$$

Since $I_{W_1,W_2}$ is an intertwining operator,

$$\begin{aligned} I_{W_1,W_2}(1_{W_1}\boxtimes \mu _{W_2})=I_{W_1,W_2}(\mu _{W_1}\boxtimes 1_{W_2})({\mathcal {R}}_{V,W_1}^{-1}\boxtimes 1_{W_2}){\mathcal {A}}_{W_1,V,W_2}; \end{aligned}$$

this leads to the composition

$$\begin{aligned} V\boxtimes (W_2\boxtimes W_1)&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{W_1,W_2}^{-1}} V\boxtimes (W_1\boxtimes W_2)\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {{\mathcal {M}}_{V,W_1}^{-1}\boxtimes 1_{W_2}} (V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {(g^{-1}\boxtimes 1_{W_1})\boxtimes 1_{W_2}} (V\boxtimes W_1)\boxtimes W_2\xrightarrow {\mu _{W_1}\boxtimes 1_{W_2}} W_1\boxtimes W_2 \\&\xrightarrow {I_{W_1,W_2}} W_1\boxtimes _V W_2. \end{aligned}$$

Since $W_1$ is a g-twisted V-module, we can eliminate $(g^{-1}\boxtimes 1_{W_1}){\mathcal {M}}_{W_1,W_2}^{-1}$ here and then add associativity and braiding isomorphisms and their inverses to obtain:

$$\begin{aligned} V\boxtimes (W_2\boxtimes W_1)&\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}} (V\boxtimes W_2)\boxtimes W_1\xrightarrow {{\mathcal {R}}_{V,W_2}\boxtimes 1_{W_1}} (W_2\boxtimes V)\boxtimes W_1 \\&\xrightarrow {{\mathcal {R}}_{V,W_2}^{-1}\boxtimes 1_{W_1}} (V\boxtimes W_2)\boxtimes W_1 \\&\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}^{-1}} V\boxtimes (W_2\boxtimes W_1)\xrightarrow {1_V\boxtimes {\mathcal {R}}_{W_1,W_2}^{-1}} V\boxtimes (W_1\boxtimes W_2) \\&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}}(V\boxtimes W_1)\boxtimes W_2 \\&\xrightarrow {\mu _{W_1}\boxtimes 1_{W_2}} W_1\boxtimes W_2\xrightarrow {I_{W_1,W_2}} W_1\boxtimes _V W_2. \end{aligned}$$

By the hexagon axiom and naturality of the braiding isomorphisms, this is

$$\begin{aligned} V\boxtimes (W_2\boxtimes W_1)&\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}} (V\boxtimes W_2)\boxtimes W_1\xrightarrow {{\mathcal {R}}_{V,W_2}\boxtimes 1_{W_1}} (W_2\boxtimes V)\boxtimes W_1 \\&\xrightarrow {{\mathcal {A}}_{W_2,V,W_1}^{-1}} W_2\boxtimes (V\boxtimes W_1) \\&\xrightarrow {1_{W_2}\boxtimes \mu _{W_1}} W_2\boxtimes W_1\xrightarrow {{\mathcal {R}}_{W_1,W_2}^{-1}} W_1\boxtimes W_2\xrightarrow {I_{W_1,W_2}} W_1\boxtimes _V W_2, \end{aligned}$$

which is the right side of (A.5). We have now proved that ${\mathcal {R}}^V_{W_1,W_2}$ exists and is an isomorphism in ${\mathcal {SC}}$.

Now we prove that ${\mathcal {R}}^V_{W_1,W_2}$ is a morphism in ${{\,\mathrm{Rep}\,}}V$ (and thus in ${{\,\mathrm{Rep}\,}}^G V$). From the commutative diagrams

and

for $i=1$ and $i=2$, together with the surjectivity of $1_V\boxtimes I_{W_1,W_2}$, it is sufficient to show

$$\begin{aligned} I_{T_g(W_2),W_1}{\mathcal {R}}_{W_1,W_2}\mu ^{(1)}_{W_1,W_2}=I_{T_g(W_2),W_1}\mu ^{(2)}_{T_g(W_2), W_1}(1_V\boxtimes {\mathcal {R}}_{W_1,W_2}). \end{aligned}$$

We start with the right side of this equation, which is the composition

$$\begin{aligned} V\boxtimes (W_1\boxtimes W_2)&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{W_1,W_2}} V\boxtimes (W_2\boxtimes W_1)\xrightarrow {{\mathcal {A}}_{V,W_2,W_1}} (V\boxtimes W_2)\boxtimes W_1 \\&\xrightarrow {{\mathcal {R}}_{V,W_2}\boxtimes 1_{W_1}} (W_2\boxtimes V)\boxtimes W_1 \\&\xrightarrow {{\mathcal {A}}_{W_2,V,W_1}^{-1}} W_2\boxtimes (V\boxtimes W_1)\xrightarrow {1_{W_2}\boxtimes \mu _{W_1}} W_2\boxtimes W_1 \\&\xrightarrow {I_{T_g(W_2),W_1}} T_g(W_2)\boxtimes _V W_1. \end{aligned}$$

By the hexagon axioms in ${\mathcal {SC}}$, this composition simplifies to

$$\begin{aligned} V\boxtimes (W_1\boxtimes W_2)&\xrightarrow {{\mathcal {A}}_{V,W_1,W_2}} (V\boxtimes W_1)\boxtimes W_2\xrightarrow {{\mathcal {R}}_{V\boxtimes W_1, W_2}} W_2\boxtimes (V\boxtimes W_1) \\&\xrightarrow {1_{W_2}\boxtimes \mu _{W_1}} W_2\boxtimes W_1\xrightarrow {I_{T_g(W_2),W_1}} T_g(W_2)\boxtimes _V W_1. \end{aligned}$$

Then we get $I_{T_g(W_2),W_1}{\mathcal {R}}_{W_1,W_2}\mu ^{(1)}_{W_1,W_2}$ from the naturality of the braiding isomorphisms.

Next we show that the ${\mathcal {R}}^V_{W_1,W_2}$ define an even natural isomorphism from $\boxtimes$ to $\boxtimes \circ (T_g\times 1_{{{\,\mathrm{Rep}\,}}^g V})\circ \sigma$, that is, for parity-homogeneous morphisms $f_1: W_1\rightarrow {\widetilde{W}}_1$ in ${{\,\mathrm{Rep}\,}}^g V$ and $f_2: W_2\rightarrow {\widetilde{W}}_2$ in ${{\,\mathrm{Rep}\,}}V$,

$$\begin{aligned} {\mathcal {R}}^V_{{\widetilde{W}}_1,{\widetilde{W}}_2}(f_1\boxtimes _V f_2) =(-1)^{\vert f_1\vert \vert f_2\vert } (T_g(f_2)\boxtimes _V f_1){\mathcal {R}}_{W_1, W_2}. \end{aligned}$$

First, ${\mathcal {R}}^V_{W_1,W_2}$ is even because ${\mathcal {R}}_{W_1,W_2}$, $I_{W_1,W_2}$, and $I_{T_g(W_2),W_1}$ are even. Then from the commutativity of

and

the surjectivity of $I_{W_1,W_2}$, and the naturality of the braiding in ${\mathcal {SC}}$, we get the naturality of ${\mathcal {R}}^V$.

To complete the proof, we need to check that the braiding ${\mathcal {R}}^V$ is compatible with the G-action and satisfies the hexagon/heptagon axioms. First, for $g,h\in G$, $W_1$ a g-twisted V-module, and $W_2$ any object in ${{\,\mathrm{Rep}\,}}V$, we need

$$\begin{aligned} \tau _{h; T_g(W_2),W_1} T_h({\mathcal {R}}^V_{W_1,W_2}) = {\mathcal {R}}^V_{T_h(W_1),T_h(W_2)} \tau _{h; W_1,W_2}. \end{aligned}$$

This follows from the commutative diagrams

and

as well as the surjectivity of $I_{W_1,W_2}$. In the second diagram here, the image of ${\mathcal {R}}^V_{T_h(W_1),T_h(W_2)}$ is indeed $T_{gh}(W_2)\boxtimes _V T_h(W_1)$: because $W_1$ is g-twisted, $T_h(W_1)$ is $hgh^{-1}$-twisted, and then $T_{hgh^{-1}}(T_h(W_2))=T_{hg}(W_2)$.

Now suppose $g_1, g_2\in G$, $W_1$ is a $g_1$-twisted V-module, $W_2$ is a $g_2$-twisted V-module, and $W_3$ is any object of ${{\,\mathrm{Rep}\,}}V$. The first hexagon axiom follows from the commutative diagrams

and

the surjectivity of $I_{W_1, W_2\boxtimes _V W_3}(1_{W_1}\boxtimes I_{W_2,W_3})$, and the hexagon axioms in ${\mathcal {SC}}$. For the heptagon, take $g\in G$, a g-twisted V-module $W_1$, and any objects $W_2$, $W_3$ in ${{\,\mathrm{Rep}\,}}V$. Then the commutative diagrams

and

the surjectivity of $I_{W_1\boxtimes _V W_2, W_3}(I_{W_1,W_2}\boxtimes 1_{W_3})$, and the hexagon axiom in ${\mathcal {SC}}$ complete the proof of the theorem.

Details for Theorem 3.3

Here we provide detailed calculations for the proofs of Sect. 3, incorporating all unit and associativity isomorphisms and making heavy use of the triangle, pentagon, and hexagon axioms.

Equations (3.1) and (3.2). We consider $e_{V\boxtimes V}(1_{V\boxtimes V}\boxtimes F_L)$, which is given by the composition

$$\begin{aligned} (V\boxtimes V) \boxtimes V&\xrightarrow {1_{V\boxtimes V}\boxtimes l_V^{-1}} (V\boxtimes V)\boxtimes ({\mathbf {1}}\boxtimes V) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes ({\widetilde{i}}_V\boxtimes 1_V)} (V\boxtimes V)\boxtimes ((V\boxtimes V)\boxtimes V) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes {\mathcal {A}}_{V,V,V}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes (1_V\boxtimes \mu _V)} (V\boxtimes V)\boxtimes (V\boxtimes V)\xrightarrow {{\mathcal {A}}^{-1}_{V,V,V\boxtimes V}} V\boxtimes (V\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,V}} V\boxtimes ((V\boxtimes V)\boxtimes V)\xrightarrow {1_V\boxtimes (\varepsilon _V\mu _V\boxtimes 1_V)} V\boxtimes ({\mathbf {1}}\boxtimes V) \\&\xrightarrow {1_V\boxtimes l_V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}. \end{aligned}$$

We move the second associativity isomorphism to the front using its naturality and we move the first $\mu _V$ back using naturality of the associativity and left unit isomorphisms:

$$\begin{aligned} (V\boxtimes V) \boxtimes V&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes (1_V\boxtimes l_V^{-1})} V\boxtimes (V\boxtimes ({\mathbf {1}}\boxtimes V)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_V))} V\boxtimes (V\boxtimes ((V\boxtimes V)\boxtimes V)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes {\mathcal {A}}_{V,V,V}^{-1})} V\boxtimes (V\boxtimes (V\boxtimes (V\boxtimes V)))\\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,V\boxtimes V}} V\boxtimes ((V\boxtimes V)\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_V\boxtimes (\varepsilon _V\mu _V\boxtimes 1_{V\boxtimes V})} V\boxtimes ({\mathbf {1}}\boxtimes (V\boxtimes V)) \xrightarrow {1_V\boxtimes l_{V\boxtimes V}} V\boxtimes (V\boxtimes V) \\&\xrightarrow {1_V\boxtimes \mu _V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}. \end{aligned}$$

We rewrite using the triangle axiom and naturality of the associativity isomorphisms:

$$\begin{aligned} (V\boxtimes V) \boxtimes V&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes (r_V^{-1}\boxtimes 1_V)} V\boxtimes ((V\boxtimes {\mathbf {1}})\boxtimes V) \\&\xrightarrow {1_V\boxtimes ((1_V\boxtimes {\widetilde{i}}_V)\boxtimes 1_V)} V\boxtimes ((V\boxtimes (V\boxtimes V))\boxtimes V) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V\boxtimes V,V}^{-1}} V\boxtimes (V\boxtimes ((V\boxtimes V)\boxtimes V)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes {\mathcal {A}}_{V,V,V}^{-1})} V\boxtimes (V\boxtimes (V\boxtimes (V\boxtimes V))) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,V\boxtimes V}} V\boxtimes ((V\boxtimes V)\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V\boxtimes V,V,V}} V\boxtimes (((V\boxtimes V)\boxtimes V)\boxtimes V) \\&\xrightarrow {1_V\boxtimes ((\varepsilon _V\mu _V\boxtimes 1_V)\boxtimes 1_V)} V\boxtimes (({\mathbf {1}}\boxtimes V)\boxtimes V) \\&\xrightarrow {1_V\boxtimes (l_V\boxtimes 1_V)} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes \mu _V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}. \end{aligned}$$

Now we replace the associativity isomorphisms in the third through sixth lines with $1_V\boxtimes ({\mathcal {A}}_{V,V,V}\boxtimes 1_V)$ using the pentagon axiom, and then by rigidity of V, the whole composition collapses to

$$\begin{aligned} (V\boxtimes V)\boxtimes V\xrightarrow {{\mathcal {A}}^{-1}_{V,V,V}} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes \mu _V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}\end{aligned}$$

(B.1)

as required.

On the other hand, $e_{V\boxtimes V}(1_{V\boxtimes V}\boxtimes F_R)$ is the composition

$$\begin{aligned} (V\boxtimes V) \boxtimes V&\xrightarrow {1_{V\boxtimes V}\boxtimes r_V^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes {\mathbf {1}})\\&\xrightarrow {1_{V\boxtimes V}\boxtimes (1_V\boxtimes {\widetilde{i}}_V)} (V\boxtimes V)\boxtimes (V\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes {\mathcal {A}}_{V,V,V}} (V\boxtimes V)\boxtimes ((V\boxtimes V)\boxtimes V) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes (\mu _V\boxtimes 1_V)} (V\boxtimes V)\boxtimes (V\boxtimes V)\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes V}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,V}} V\boxtimes ((V\boxtimes V)\boxtimes V) \xrightarrow {1_V\boxtimes (\varepsilon _V\mu _V\boxtimes 1_V)} V\boxtimes ({\mathbf {1}}\boxtimes V) \\&\xrightarrow {1_V\boxtimes l_V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V}{\mathbf {1}}. \end{aligned}$$

As before, we move ${\mathcal {A}}_{V,V,V\boxtimes V}^{-1}$ forward and the first $\mu _V$ back; we also apply the associativity of $\mu _V$:

$$\begin{aligned} (V\boxtimes V) \boxtimes V&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes (1_V\boxtimes r_V^{-1})} V\boxtimes (V\boxtimes (V\boxtimes {\mathbf {1}}))\\&\xrightarrow {1_V\boxtimes (1_V\boxtimes (1_V\boxtimes {\widetilde{i}}_V))} V\boxtimes (V\boxtimes (V\boxtimes (V\boxtimes V))) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes {\mathcal {A}}_{V,V,V})} V\boxtimes (V\boxtimes ((V\boxtimes V)\boxtimes V)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V\boxtimes V,V}} V\boxtimes ((V\boxtimes (V\boxtimes V))\boxtimes V) \\&\xrightarrow {1_V\boxtimes ({\mathcal {A}}_{V,V,V}\boxtimes 1_V)} V\boxtimes (((V\boxtimes V)\boxtimes V)\boxtimes V) \\&\xrightarrow {1_V\boxtimes ((\mu _V\boxtimes 1_V)\boxtimes 1_V)} V\boxtimes ((V\boxtimes V)\boxtimes V) \\&\xrightarrow {1_V\boxtimes (\varepsilon _V\mu _V\boxtimes 1_V)} V\boxtimes ({\mathbf {1}}\boxtimes V) \xrightarrow {1_V\boxtimes l_V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}. \end{aligned}$$

Now we rewrite the associativity isomorphisms in the third through fifth rows as ${\mathcal {A}}_{V\boxtimes V,V,V}{\mathcal {A}}_{V,V,V\boxtimes V}$ using the pentagon axiom, and then we apply the naturality of these isomorphisms:

$$\begin{aligned} (V \boxtimes V) \boxtimes V&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes (1_V\boxtimes r_V^{-1})} V\boxtimes (V\boxtimes (V\boxtimes {\mathbf {1}}))\\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,{\mathbf {1}}}} V\boxtimes ((V\boxtimes V)\boxtimes {\mathbf {1}}) \\&\xrightarrow {1_V\boxtimes (1_{V\boxtimes V}\boxtimes {\widetilde{i}}_V)} V\boxtimes ((V\boxtimes V)\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_V\boxtimes (\mu _V\boxtimes 1_{V\boxtimes V})} V\boxtimes (V\boxtimes (V\boxtimes V)) \xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,V}} V\boxtimes ((V\boxtimes V)\boxtimes V) \\&\xrightarrow {1_V\boxtimes (\varepsilon _V\mu _V\boxtimes 1_V)} V\boxtimes ({\mathbf {1}}\boxtimes V)\xrightarrow {1_V\boxtimes l_V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}. \end{aligned}$$

Next we use the identity ${\mathcal {A}}_{V,V,{\mathbf {1}}}(1_V\boxtimes r_V^{-1}) =r_{V\boxtimes V}^{-1}$ and the naturality of the right unit isomorphisms:

$$\begin{aligned} (V \boxtimes V) \boxtimes V&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes \mu _V} V\boxtimes V\xrightarrow {1_V\boxtimes r_V^{-1}} V\boxtimes (V\boxtimes {\mathbf {1}}) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes {\widetilde{i}}_V)} V\boxtimes (V\boxtimes (V\boxtimes V)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,V}} V\boxtimes ((V\boxtimes V)\boxtimes V)\xrightarrow {1_V\boxtimes (\varepsilon _V\mu _V\boxtimes 1_V)} V\boxtimes ({\mathbf {1}}\boxtimes V) \\&\xrightarrow {1_V\boxtimes l_V} V\boxtimes V\xrightarrow {\varepsilon _V\mu _V} {\mathbf {1}}. \end{aligned}$$

Finally, this composition collapses to (B.1) by the rigidity of V.

Equation (3.3). By the left unit property of V, $(\mathrm {Tr}_{\mathcal {C}}\,g)1_V$ is the composition

$$\begin{aligned} V\xrightarrow {l_V^{-1}}{\mathbf {1}}\boxtimes V\xrightarrow {{\widetilde{i}}_V\boxtimes 1_V} (V\boxtimes V)\boxtimes V\xrightarrow {(1_V\boxtimes g)\boxtimes 1_V} (V\boxtimes V)\boxtimes V\xrightarrow {\mu _V\boxtimes 1_V} V\boxtimes V\xrightarrow {\mu _V} V. \end{aligned}$$

Because g is an automorphism of V, this agrees with

$$\begin{aligned} \begin{aligned} V&\xrightarrow {l_V^{-1}} {\mathbf {1}}\boxtimes V\xrightarrow {{\widetilde{i}}_V\boxtimes 1_V} (V\boxtimes V)\boxtimes V\xrightarrow {(g^{-1}\boxtimes 1_V)\boxtimes g^{-1}} (V\boxtimes V)\boxtimes V \\&\xrightarrow {\mu _V\boxtimes 1_V} V\boxtimes V\xrightarrow {\mu _V} V\xrightarrow {g} V. \end{aligned} \end{aligned}$$

We then use associativity of $\mu _V$ and naturality of associativity and unit isomorphisms to rewrite as

$$\begin{aligned} \begin{aligned} V&\xrightarrow {g^{-1}} V\xrightarrow {l_V^{-1}} {\mathbf {1}}\boxtimes V\xrightarrow {{\widetilde{i}}_V\boxtimes 1_V} (V\boxtimes V)\boxtimes V\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}} V\boxtimes (V\boxtimes V) \\&\xrightarrow {1_V\boxtimes \mu _V} V\boxtimes V\xrightarrow {g^{-1}\boxtimes 1_V} V\boxtimes V\xrightarrow {\mu _V} V\xrightarrow {g} V. \end{aligned} \end{aligned}$$

Next we use Lemma 3.7 and the automorphism property of g to obtain

$$\begin{aligned} \begin{aligned} V&\xrightarrow {g^{-1}} V\xrightarrow {r_V^{-1}} V\boxtimes {\mathbf {1}}\xrightarrow {1_V\boxtimes {\widetilde{i}}_V} V\boxtimes (V\boxtimes V)\xrightarrow {{\mathcal {A}}_{V,V,V}} (V\boxtimes V)\boxtimes V \\&\xrightarrow {\mu _V\boxtimes 1_V} V\boxtimes V\xrightarrow {1_V\boxtimes g} V\boxtimes V\xrightarrow {\mu _V} V. \end{aligned} \end{aligned}$$

Naturality of the associativity isomorphisms and one more application of the associativity of $\mu _V$ then yields

$$\begin{aligned} \begin{aligned} V&\xrightarrow {g^{-1}} V\xrightarrow {r_V^{-1}} V\boxtimes {\mathbf {1}}\xrightarrow {1_V\boxtimes {\widetilde{i}}_V} V\boxtimes (V\boxtimes V)\xrightarrow {1_V\boxtimes (1_V\boxtimes g)} V\boxtimes (V\boxtimes V) \\&\xrightarrow {1_V\boxtimes \mu _V} V\boxtimes V\xrightarrow {\mu _V} V, \end{aligned} \end{aligned}$$

which is $(\mathrm {Tr}_{\mathcal {C}}\,g)g^{-1}$ by the right unit property of V.

Equation (3.4). We start with $\mu _W(1_V\boxtimes \Pi _g)$, which is the composition

$$\begin{aligned} V\boxtimes W&\xrightarrow {1_V\boxtimes l_W^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}])} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes \mu _W)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

We replace the first arrow using the triangle axiom and the triviality of ${\mathcal {R}}_{{\mathbf {1}},V}$ and rewrite the last two arrows using associativity of $\mu _V$ and $\mu _W$:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_V^{-1}\boxtimes 1_W} ({\mathbf {1}}\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {R}}_{{\mathbf {1}},V}\boxtimes 1_W} (V\boxtimes {\mathbf {1}})\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,{\mathbf {1}},W}^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}])} V\boxtimes (V\boxtimes (V\boxtimes W))\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes (\mu _V\boxtimes 1_W)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Now we write $l_V^{-1}\boxtimes 1_W={\mathcal {A}}_{{\mathbf {1}},V,W} l_{V\boxtimes W}^{-1}$ and apply naturality of associativity and braiding isomorphisms to ${\widetilde{i}}_V$; meanwhile we rewrite the last three arrows using associativity again and naturality of the associativity isomorphisms:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {R}}_{V\boxtimes V,V}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V\boxtimes V,W}^{-1}} V\boxtimes ((V\boxtimes V)\boxtimes W)\\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}])} V\boxtimes (V\boxtimes (V\boxtimes W))\\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \xrightarrow {{\mathcal {A}}_{V,V\boxtimes V,W}} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {(1_V\boxtimes \mu _V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Next we apply the hexagon and pentagon axioms to the arrows in the third and fourth lines above; we also apply the associativity $\mu _V$ and the pentagon axiom towards the end of the composition:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {({\mathcal {R}}_{V,V}\boxtimes 1_V)\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes W)\\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}])} V\boxtimes (V\boxtimes (V\boxtimes W))\\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {(\mu _V\boxtimes 1_V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

We use naturality of the associativity isomorphisms to cancel ${\mathcal {A}}_{V,V,V\boxtimes W}$ and its inverse here. With this done, we move the second ${\mathcal {R}}_{V,V}$ using naturality of the associativity isomorphisms, in order to cancel it against the first $\mu _V$ using commutativity of $\mu _V$. Then we begin rewriting the last line using associativity again:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes W)\xrightarrow {1_{V\boxtimes V}\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}]} (V\boxtimes V)\boxtimes (V\boxtimes W)\\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes \mu _V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\\&\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

We now rewrite the last five arrows using associativity of $\mu _W$, naturality of the associativity isomorphisms, and the pentagon axiom. Then we use commutativity to insert an ${\mathcal {R}}_{V,V}$ in front of $\mu _V$:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes W)\xrightarrow {1_{V\boxtimes V}\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}]} (V\boxtimes V)\boxtimes (V\boxtimes W)\\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W)\xrightarrow {1_V\boxtimes ({\mathcal {R}}_{V,V}\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W)\\&\xrightarrow {1_V\boxtimes (\mu _V\boxtimes 1_W)} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Next we use naturality of the associativity isomorphisms to move ${\mathcal {A}}_{V,V,V\boxtimes W}^{-1}$, the pentagon axiom in the fourth and fifth lines, the associativity of $\mu _W$, and naturality of the associativity and braiding isomorphisms to move g:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V\boxtimes V,W}^{-1}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W))\xrightarrow {1_V\boxtimes (1_V\boxtimes {\mathcal {M}}_{V,W})} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes ({\mathcal {R}}_{V,V}\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W)\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes \mu _W)} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes (g\boxtimes 1_W)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Now we use naturality of the associativity isomorphisms to move the first ${\mathcal {R}}_{V,V}$, and then we use the pentagon axiom to rewrite the first three associativity isomorphisms and the hexagon axiom to rewrite all braiding isomorphisms:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W)\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{V,V\boxtimes W}} V\boxtimes ((V\boxtimes W)\boxtimes V) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,W,V}^{-1}} V\boxtimes (V\boxtimes (W\boxtimes V))\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,W,V}} V\boxtimes ((V\boxtimes W)\boxtimes V) \\&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{V\boxtimes W,V}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W)\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes \mu _W)} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes (g\boxtimes 1_W)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

We cancel all pairs of associativity isomorphisms and their inverses and then apply naturality of the associativity, braiding, and unit isomorphisms to the first $\mu _W$ to finally obtain

$$\begin{aligned} V\boxtimes W&\xrightarrow {\mu _W} W\xrightarrow {l_W^{-1}}{\mathbf {1}}\boxtimes W \xrightarrow {{\widetilde{i}}_V\boxtimes 1_W} (V\boxtimes V)\boxtimes W \xrightarrow {{\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {M}}_{V,W}} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes (g\boxtimes 1_W)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W, \end{aligned}$$

which is $\Pi _g\mu _W$.

Equations (3.5) through (3.7). The morphism $\mu _W(g\boxtimes \Pi _g){\mathcal {M}}_{V,W}$ is the composition

$$\begin{aligned} V\boxtimes W&\xrightarrow {{\mathcal {M}}_{V,W}} V\boxtimes W\xrightarrow {1_V\boxtimes l_W^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W) \nonumber \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W))\nonumber \\&\xrightarrow {g\boxtimes (1_V\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}])} V\boxtimes (V\boxtimes (V\boxtimes W))\xrightarrow {1_V\boxtimes (1_V\boxtimes \mu _W)} V\boxtimes (V\boxtimes W) \nonumber \\&\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

(B.2)

We begin by rewriting the second, third, and fourth arrows:

$$\begin{aligned}&(1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}) (1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_W)){\mathcal {A}}_{V,{\mathbf {1}},W}^{-1}(r_V^{-1}\boxtimes 1_W)\nonumber \\&\quad =(1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}){\mathcal {A}}_{V,V\boxtimes V, W}^{-1}((1_V\boxtimes {\widetilde{i}}_V)\boxtimes 1_W)({\mathcal {R}}_{{\mathbf {1}}, V}\boxtimes 1_W)(l_V^{-1}\boxtimes 1_W)\nonumber \\&\quad =(1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}){\mathcal {A}}_{V,V\boxtimes V, W}^{-1}({\mathcal {R}}_{V\boxtimes V, V}\boxtimes 1_W)(({\widetilde{i}}_V\boxtimes 1_V)\boxtimes 1_W){\mathcal {A}}_{{\mathbf {1}},V,W} l_{V\boxtimes W}^{-1}\nonumber \\&\quad ={\mathcal {A}}_{V,V,V\boxtimes W}^{-1}{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}(({\mathcal {R}}_{V,V}\boxtimes 1_V)\boxtimes 1_W)({\mathcal {A}}_{V,V,V}\boxtimes 1_W)((1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W)\circ \nonumber \\&\qquad \circ ({\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W){\mathcal {A}}_{V\boxtimes V,V,W}({\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}) l_{V\boxtimes W}^{-1}, \end{aligned}$$

(B.3)

where the last equality uses both the hexagon and pentagon axioms. We also rewrite the last three arrows of (B.2) using the associativity and commutativity of $\mu _W$ and $\mu _V$ as well as the pentagon axiom:

$$\begin{aligned}&\mu _W(1_V \boxtimes \mu _W) (1_V\boxtimes (1_V\boxtimes \mu _W)) \nonumber \\&\quad = \mu _W(1_V\boxtimes \mu _W)(1_V\boxtimes (\mu _V\boxtimes 1_W))(1_V\boxtimes {\mathcal {A}}_{V,V,W})\nonumber \\&\quad =\mu _W(\mu _V\boxtimes 1_W){\mathcal {A}}_{V,V,W}(1_V\boxtimes (\mu _V\boxtimes 1_W))(1_V\boxtimes ({\mathcal {R}}_{V,V}\boxtimes 1_W))(1_V\boxtimes {\mathcal {A}}_{V,V,W})\nonumber \\&\quad =\mu _W(\mu _V\boxtimes 1_W)((1_V\boxtimes \mu _V)\boxtimes 1_W)((1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W){\mathcal {A}}_{V,V\boxtimes V,W}(1_V\boxtimes {\mathcal {A}}_{V,V,W})\nonumber \\&\quad =\mu _W(\mu _V\boxtimes 1_W)((\mu _V\boxtimes 1_V)\boxtimes 1_W)({\mathcal {A}}_{V,V,V}\boxtimes 1_W) \nonumber \\&\qquad ((1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W)({\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W){\mathcal {A}}_{V\boxtimes V,V,W}{\mathcal {A}}_{V,V,V\boxtimes W}. \end{aligned}$$

(B.4)

We insert (B.3) and (B.4) into (B.2), canceling ${\mathcal {A}}_{V,V,V\boxtimes W}$ with its inverse:

$$\begin{aligned} V \boxtimes W&\xrightarrow {{\mathcal {M}}_{V,W}} V\boxtimes W\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V, V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {({\mathcal {R}}_{V,V}\boxtimes 1_V)\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {(g\boxtimes 1_V)\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}]} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {(\mu _V\boxtimes 1_V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W \\&\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Next we apply naturality of the left unit isomorphisms to the first two arrows and naturality of the associativity and braiding isomorphisms to g. Then we use the automorphism property of g and finally apply naturality of associativity to the second ${\mathcal {R}}_{V,V}$:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \nonumber \\&\xrightarrow {1_{V\boxtimes V}\boxtimes {\mathcal {M}}_{V,W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\nonumber \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V, V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \nonumber \\&\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\nonumber \\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes W) \nonumber \\&\xrightarrow {1_{V\boxtimes V}\boxtimes {\mathcal {M}}_{V,W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\nonumber \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {({\mathcal {R}}_{V,V}\boxtimes 1_V)\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \nonumber \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\nonumber \\&\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \nonumber \\&\xrightarrow {(\mu _V\boxtimes 1_V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\nonumber \\&\xrightarrow {(g\boxtimes 1_V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

(B.5)

Now we use the hexagon axiom and commutativity of $\mu _V$ to simplify the penultimate seven arrows:

$$\begin{aligned}&\mu _V(g\boxtimes 1_V)(\mu _V\boxtimes 1_V){\mathcal {A}}_{V,V,V}(1_V\boxtimes {\mathcal {R}}_{V,V}) {\mathcal {A}}_{V,V,V}^{-1}({\mathcal {R}}_{V,V}\boxtimes 1_V) \\&\quad = \mu _V(g\boxtimes 1_V)(\mu _V\boxtimes 1_V){\mathcal {R}}_{V,V\boxtimes V}{\mathcal {A}}_{V,V,V}^{-1} \\&\quad =\mu _V{\mathcal {R}}_{V,V}(1_V\boxtimes g)(1_V\boxtimes \mu _V){\mathcal {A}}_{V,V,V}^{-1} =\mu _V(1_V\boxtimes g)(1_V\boxtimes \mu _V){\mathcal {A}}_{V,V,V}^{-1}. \end{aligned}$$

We also rewrite associativity isomorphisms in the third and fifth lines of (B.5) using the pentagon axiom:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes {\mathcal {M}}_{V,W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V,V\boxtimes V,W}} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {(1_V\boxtimes {\mathcal {R}}_{V,V})\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V\boxtimes V, W}^{-1}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\xrightarrow {1_{V\boxtimes V}\boxtimes {\mathcal {M}}_{V,W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes \mu _V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W \\&\xrightarrow {(1_V\boxtimes g)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Again using naturality of associativity and the pentagon axiom, we get:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \nonumber \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W))\nonumber \\&\xrightarrow {1_V\boxtimes (1_V\boxtimes {\mathcal {M}}_{V,W})} V\boxtimes (V\boxtimes (V\boxtimes W))\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \nonumber \\&\xrightarrow {1_V\boxtimes ({\mathcal {R}}_{V,V}\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W)\nonumber \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W))\xrightarrow {1_V\boxtimes (1_V\boxtimes {\mathcal {M}}_{V,W})} V\boxtimes (V\boxtimes (V\boxtimes W)) \nonumber \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W)\nonumber \\&\xrightarrow {{\mathcal {A}}_{V,V\boxtimes V,W}} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes \mu _V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W \nonumber \\&\xrightarrow {(1_V\boxtimes g)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

(B.6)

We now analyze the isomorphism $V\boxtimes (V\boxtimes W)\rightarrow (V\boxtimes V)\boxtimes W$ in the fourth through ninth arrows. We apply the Yang–Baxter relation once to get

$$\begin{aligned} V\boxtimes (V\boxtimes W)&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{V,W}} V\boxtimes (W\boxtimes V)\xrightarrow {{\mathcal {A}}_{V,W,V}} (V\boxtimes W)\boxtimes V \\&\xrightarrow {{\mathcal {R}}_{V,W}\boxtimes 1_V} (W\boxtimes V)\boxtimes V\xrightarrow {{\mathcal {A}}_{W,V,V}^{-1}} W\boxtimes (V\boxtimes V) \\&\xrightarrow {1_W\boxtimes {\mathcal {R}}_{V,V}} W\boxtimes (V\boxtimes V)\xrightarrow {{\mathcal {A}}_{W,V,V}} (W\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {R}}_{W,V}\boxtimes 1_V} (V\boxtimes W)\boxtimes V\xrightarrow {{\mathcal {A}}_{V,W,V}^{-1}} V\boxtimes (W\boxtimes V) \\&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{W,V}} V\boxtimes (V\boxtimes W)\xrightarrow {{\mathcal {A}}_{V,V,W}} (V\boxtimes V)\boxtimes W \end{aligned}$$

and a second time to obtain

$$\begin{aligned} V\boxtimes (V\boxtimes W)&\xrightarrow {1_V\boxtimes {\mathcal {R}}_{V,W}} V\boxtimes (W\boxtimes V)\xrightarrow {{\mathcal {A}}_{V,W,V}} (V\boxtimes W)\boxtimes V \\&\xrightarrow {{\mathcal {R}}_{V,W}\boxtimes 1_V} (W\boxtimes V)\boxtimes V\xrightarrow {{\mathcal {R}}_{W,V}\boxtimes 1_V} (V\boxtimes W)\boxtimes V \\&\xrightarrow {{\mathcal {A}}_{V,W,V}^{-1}} V\boxtimes (W\boxtimes V)\xrightarrow {1_V\boxtimes {\mathcal {R}}_{W,V}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V,V,W}} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {R}}_{V,V}\boxtimes 1_W} (V\boxtimes V)\boxtimes W. \end{aligned}$$

By the hexagon axiom, this equals

$$\begin{aligned} V\boxtimes (V\boxtimes W)&\xrightarrow {{\mathcal {A}}_{V,V,W}} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {R}}_{V\boxtimes V, W}} W\boxtimes (V\boxtimes V)\xrightarrow {{\mathcal {A}}_{W,V,V}}(W\boxtimes V)\boxtimes V \\&\xrightarrow {{\mathcal {A}}_{W,V,V}^{-1}} W\boxtimes (V\boxtimes V)\xrightarrow {{\mathcal {R}}_{W,V\boxtimes V}} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {R}}_{V,V}\boxtimes 1_W} (V\boxtimes V)\boxtimes W. \end{aligned}$$

We cancel associativity isomorphisms and insert this composition into (B.6):

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_{V\boxtimes W}^{-1}} {\mathbf {1}}\boxtimes (V\boxtimes W)\xrightarrow {{\widetilde{i}}_V\boxtimes 1_{V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W)\xrightarrow {1_V\boxtimes {\mathcal {M}}_{V\boxtimes V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes ({\mathcal {R}}_{V,V}\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V\boxtimes V,W}} (V\boxtimes (V\boxtimes V))\boxtimes W\xrightarrow {(1_V\boxtimes \mu _V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W \\&\xrightarrow {(1_V\boxtimes g)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

By the naturality of associativity and monodromy, the commutativity of $\mu _V$, and properties of the left unit isomorphism, we now get

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_V^{-1}\boxtimes 1_W} ({\mathbf {1}}\boxtimes V)\boxtimes W\xrightarrow {({\widetilde{i}}_V\boxtimes 1_V)\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V\boxtimes V,V,W}^{-1}} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes (\mu _V\boxtimes 1_W)} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {M}}_{V,W}} V\boxtimes (V\boxtimes W)\xrightarrow {{\mathcal {A}}_{V,V,W}} (V\boxtimes V)\boxtimes W \\&\xrightarrow {(1_V\boxtimes g)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

By the pentagon axiom, the third through fifth arrows equal ${\mathcal {A}}_{V,V\boxtimes V, W}^{-1}({\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W)$. We also use naturality of the associativity isomorphisms and associativity of $\mu _V$ to end up with:

$$\begin{aligned} V \boxtimes W&\xrightarrow {l_V^{-1}\boxtimes 1_W} ({\mathbf {1}}\boxtimes V)\boxtimes W\xrightarrow {({\widetilde{i}}_V\boxtimes 1_V)\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}^{-1}\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {(1_V\boxtimes \mu _V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_{V}\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}]} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

At this point, we use Lemma 3.7 to replace the morphism $V\rightarrow V\boxtimes V$ in the first four arrows with

$$\begin{aligned} (\mu _V\boxtimes 1_V){\mathcal {A}}_{V,V,V}(1_V\boxtimes {\widetilde{i}}_V)r_V^{-1}. \end{aligned}$$

Inserting this into the above composition and using the triangle axiom, we get

$$\begin{aligned} V \boxtimes W&\xrightarrow {1_V\boxtimes l_W^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {{\mathcal {A}}_{V,{\mathbf {1}}, W}} (V\boxtimes {\mathbf {1}})\boxtimes W\xrightarrow {(1_V\boxtimes {\widetilde{i}}_V)\boxtimes 1_W} (V\boxtimes (V\boxtimes V))\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,V}\boxtimes 1_W} ((V\boxtimes V)\boxtimes V)\boxtimes W\xrightarrow {(\mu _V\boxtimes 1_V)\boxtimes 1_W} (V\boxtimes V)\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_{V}\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}]} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Using naturality of the associativity and the pentagon, we obtain

$$\begin{aligned} V \boxtimes W&\xrightarrow {1_V\boxtimes l_W^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {{\mathcal {A}}_{V,V,V\boxtimes W}} (V\boxtimes V)\boxtimes (V\boxtimes W)\xrightarrow {1_{V\boxtimes V}\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}]} (V\boxtimes V)\boxtimes (V\boxtimes W) \\&\xrightarrow {1_{V\boxtimes V}\boxtimes \mu _W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Finally, by naturality of the associativity isomorphisms and the associativity of $\mu _V$ and $\mu _W$, this equals

$$\begin{aligned} V \boxtimes W&\xrightarrow {1_V\boxtimes l_W^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes ({\widetilde{i}}_V\boxtimes 1_W)} V\boxtimes ((V\boxtimes V)\boxtimes W) \\&\xrightarrow {1_V\boxtimes {\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes (V\boxtimes W)) \\&\xrightarrow {1_{V}\boxtimes (1_V\boxtimes [(g\boxtimes 1_W){\mathcal {M}}_{V,W}])} V\boxtimes (V\boxtimes (V\boxtimes W))\xrightarrow {1_V\boxtimes (1_V\boxtimes \mu _W)} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W, \end{aligned}$$

which is $\mu _W(1_V\boxtimes \Pi _g)$.

Equations (3.8) and (3.9). When W is h-twisted, $\Pi _g$ is the composition

$$\begin{aligned} \begin{aligned} W&\xrightarrow {l_W^{-1}} {\mathbf {1}}\boxtimes W\xrightarrow {{\widetilde{i}}_V\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes (h^{-1}g\boxtimes 1_W)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned} \end{aligned}$$

Naturality of the associativity isomorphisms and associativity of $\mu _W$ imply this is

$$\begin{aligned} \begin{aligned} W&\xrightarrow {l_W^{-1}} {\mathbf {1}}\boxtimes W\xrightarrow {{\widetilde{i}}_V\boxtimes 1_W} (V\boxtimes V)\boxtimes W \\&\xrightarrow {(1_V\boxtimes h^{-1}g)\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {\mu _V\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W, \end{aligned} \end{aligned}$$

which is the right side of (3.8). Finally, $\sum _{g\in G}\pi _g$ is the composition

$$\begin{aligned} W&\xrightarrow {l_W^{-1}}{\mathbf {1}}\boxtimes W\xrightarrow {{\widetilde{i}}_V\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes ( V\boxtimes W) \xrightarrow {1_V\boxtimes {\mathcal {M}}_{V,W}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes (\frac{1}{\vert G\vert }\sum _{g\in G} g\boxtimes 1_V)} V\boxtimes (V\boxtimes W)\xrightarrow {1_V\boxtimes \mu _W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Using $\frac{1}{\vert G\vert }\sum _{g\in G} g=\iota _V\varepsilon _V$ and the unit property of W, we get

$$\begin{aligned} W&\xrightarrow {l_W^{-1}}{\mathbf {1}}\boxtimes W\xrightarrow {{\widetilde{i}}_V\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {{\mathcal {A}}_{V,V,W}^{-1}} V\boxtimes ( V\boxtimes W) \xrightarrow {1_V\boxtimes {\mathcal {M}}_{V,W}} V\boxtimes (V\boxtimes W) \\&\xrightarrow {1_V\boxtimes (\varepsilon _V\boxtimes 1_W)} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes l_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

Then we simplify using naturality of the monodromy and associativity isomorphisms together with ${\mathcal {M}}_{{\mathbf {1}},W}=1_{{\mathbf {1}},W}$:

$$\begin{aligned} \begin{aligned} W&\xrightarrow {l_W^{-1}}{\mathbf {1}}\boxtimes W\xrightarrow {{\widetilde{i}}_V\boxtimes 1_W} (V\boxtimes V)\boxtimes W\xrightarrow {(1_V\boxtimes \varepsilon _V)\boxtimes 1_W} )(V\boxtimes {\mathbf {1}})\boxtimes W \\&\xrightarrow {{\mathcal {A}}_{V,{\mathbf {1}},W}^{-1}} V\boxtimes ({\mathbf {1}}\boxtimes W)\xrightarrow {1_V\boxtimes l_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned} \end{aligned}$$

Now $(1_V\boxtimes l_W){\mathcal {A}}_{V,{\mathbf {1}},W}^{-1} = r_V\boxtimes 1_W$ by the triangle axiom and we get

$$\begin{aligned} W\xrightarrow {l_W^{-1}}{\mathbf {1}}\boxtimes W\xrightarrow {[r_V(1_V\boxtimes \varepsilon _V){\widetilde{i}}_V]\boxtimes 1_W} V\boxtimes W\xrightarrow {\mu _W} W. \end{aligned}$$

This is the right side of (3.9) by the unit property of W.