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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 $\A\x\Mod$.

Eilenberg-Watts Theorem for 2-categories and quasi-monoidal structures for module categories over bialgebroid categories