This paper concerns the problem of lifting a KZ doctrine P to the 2-category of pseudo T-algebras for some pseudomonad T. Here we show that this problem is equivalent to giving a pseudo-distributive law (meaning that the lifted pseudomonad is automatically KZ), and that such distributive laws may be simply described algebraically and are essentially unique [as known to be the case in the (co)KZ over KZ setting]. Moreover, we give a simple description of these distributive laws using Bunge and Funk’s notion of admissible morphisms for a KZ doctrine (the principal goal of this paper). We then go on to show that the 2-category of KZ doctrines on a 2-category is biequivalent to a poset. We will also discuss here the problem of lifting a locally fully faithful KZ doctrine, which we noted earlier enjoys most of the axioms of a Yoneda structure, and show that a bijection between oplax and lax structures is exhibited on the lifted “Yoneda structure” similar to Kelly’s doctrinal adjunction. We also briefly discuss how this bijection may be viewed as a coherence result for oplax functors out of the bicategories of spans and polynomials, but leave the details for a future paper.

中文翻译：

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通过可受理性的分配法

本文涉及将一个 KZ 学说 P 提升到一些伪单子 T 的伪 T-代数的 2-范畴的问题。 这里我们证明这个问题等价于给出一个伪分配律（意思是提升的伪单子自动为 KZ )，并且这种分配律可以简单地用代数来描述并且本质上是唯一的 [正如已知的 (co)KZ over KZ 设置中的情况]。此外，我们使用 Bunge 和 Funk 的 KZ 学说的可容许态射概念（本文的主要目标）对这些分配律进行了简单的描述。然后我们继续证明 2-category 上的 KZ 学说的 2-category 与偏序集是双等价的。我们还将在这里讨论解除局部完全忠实的 KZ 学说的问题，我们之前注意到它享有米田结构的大部分公理，并表明在类似于凯利的教义附属物的提升的“米田结构”上展示了 oplax 和松散结构之间的双射。我们还简要讨论了如何将这种双射视为 oplax 函子在跨度和多项式双范畴之外的相干结果，但将详细信息留给以后的论文。