Let ${𝒞}$ be small category and ${𝒜}$ an arbitrary category. Consider the category ${𝒞}({𝒜})$ whose objects are functors from ${𝒞}$ to ${𝒜}$ and whose morphisms are natural transformations. Let ${ℬ}$ be another category, and again, consider the category ${𝒞}({ℬ})$. Now, given a functor $F:{𝒜}\to {ℬ}$ we construct the induced functor $F_{{𝒞}}:{𝒞}({𝒜})\to {𝒞}({ℬ})$. Assuming ${𝒜}$ and ${ℬ}$ to be abelian categories, it follows that the categories ${𝒞}({𝒜})$ and ${𝒞}({ℬ})$ are also abelian. We have two main goals: first, to find a relationship between the derived category $D({𝒞}({𝒜}))$ and the category ${𝒞}(D({𝒜}))$; second relate the functors $R(F_{{𝒞}})$ and ${(RF)}_{{𝒞}}:{𝒞}(D({𝒜}))\to {𝒞}(D({ℬ}))$. We apply the general results obtained to the special case of quiver sheaves.

让${𝒞}$是小类别和${𝒜}$任意类别。考虑类别${𝒞}({𝒜})$其对象是函子${𝒞}$至${𝒜}$并且其态射是自然变换。让${ℬ}$是另一个类别，再次考虑该类别${𝒞}({ℬ})$. 现在，给定一个函子$F：{𝒜}\to {ℬ}$我们构造诱导函子$F_{{𝒞}}：{𝒞}({𝒜})\to {𝒞}({ℬ})$. 假设${𝒜}$和${ℬ}$为阿贝尔范畴，因此范畴${𝒞}({𝒜})$和${𝒞}({ℬ})$也是阿贝尔。我们有两个主要目标：首先，找到派生类别之间的关系$D({𝒞}({𝒜}))$和类别${𝒞}(D({𝒜}))$; 第二个关联函子$R(F_{{𝒞}})$和${(RF)}_{{𝒞}}：{𝒞}(D({𝒜}))\to {𝒞}(D({ℬ}))$. 我们将获得的一般结果应用于颤动滑轮的特殊情况。