We prove Eilenberg-Watts Theorem for 2-categories of the representationcategories $\C\x\Mod$ of finite tensor categories $\C$. For a consequence weobtain that any autoequivalence of $\C\x\Mod$ is given by tensoring with arepresentative of some class in the Brauer-Picard group $\BrPic(\C)$. Weintroduce bialgebroid categories over $\C$ and a cohomology over a symmetricbialgebroid category. This cohomology turns out to be a generalization of theone we developed in a previous paper and moreover, an analogousVillamayor-Zelinsky sequence exists in this setting. In this context, for asymmetric bialgebroid category $\A$, we interpret the middle cohomology groupappearing in the third level of the latter sequence. We obtain a group ofquasi-monoidal structures on the representation category $\A\x\Mod$.
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机译：我们证明有限张量类别$ \ C $的表示类别$ \ C \ x \ Mod $的2类的Eilenberg-Watts定理。结果，我们得到了$ \ C \ x \ Mod $的任何自等价性，是通过在Brauer-Picard组$ \ BrPic（\ C）$中用某个类的代表表示张量给出的。我们引入了$ \ C $以上的双代数类和对称双数类的同调性。事实证明，这种同调是我们先前论文中开发的同化，并且在这种情况下存在类似的Villamayor-Zelinsky序列。在这种情况下，对于非对称双代数类别$ \ A $，我们解释出现在后一个序列的第三层的中间同调群。我们在表示类别$ \ A \ x \ Mod $上获得了一组准Monoidal结构。
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