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

We prove Eilenberg-Watts Theorem for 2-categories of the representation categories C-Mod of finite tensor categories C. For a consequence we obtain that any autoequivalence of C-Mod is given by tensoring with a representative of some class in the Brauer-Picard group BrPic(C). We introduce bialgebroid categories over C and a cohomology over a symmetric bialgebroid category. This cohomology turns out to be a generalization of the one we developed in a previous paper and moreover, an analogous Villamayor-Zelinsky sequence exists in this setting. In this context, for a symmetric bialgebroid category A, we interpret the middle cohomology group appearing in the third level of the latter sequence. We obtain a group of quasi-monoidal structures on the representation category A-Mod.