Bimodule categories realized as internal bimodules

Question

Let $\mathcal C$ be a finite tensor category, and $\mathcal M$ a finite left $\mathcal C$-module category. By a result of P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik (http://www-math.mit.edu/~etingof/tenscat1.pdf , Thm. 2.11.6), there is an algebra object $A \in \mathcal C$ and an equivalence of left $\mathcal C$-module categories $\mathcal M \simeq \mathsf{Mod}\mbox{-}A(\mathcal C)$, where $\mathsf{Mod}\mbox{-}A(\mathcal C)$ is the category of right $A$-modules internal in $\mathcal C$. There is a similar result for right $\mathcal C$-module categories.

Now assume that $\mathcal M$ is a $\mathcal C$-$\mathcal C$-bimodule category, e.g. $\mathcal M$ is braided (then there is a canonical right/left $\mathcal{C}$-module structure if $\mathcal M$ was a left/right $\mathcal{C}$-module). (Of course, one can also consider the more general case of a $\mathcal C$-$\mathcal D$-bimodule category.) What is the corresponding result to the one above? Something straight-forward would be the following: If $\mathcal M \simeq \mathsf{Mod}\mbox{-}A(\mathcal C)$ as left module categories and $\mathcal M \simeq B\mbox{-}\mathsf{Mod}(\mathcal C)$ as right module categories, then $\mathcal M \simeq B\mbox{-}\mathsf{Mod}\mbox{-}A(\mathcal C)$ as bimodule categories, where $B\mbox{-}\mathsf{Mod}\mbox{-}A(\mathcal C)$ is the category of $B$-$A$-bimodules in $\mathcal C$. If this is correct, how can it be proven? Is there already a paper covering this?

(By results of Douglas, Schommer-Pries and Snyder, it at least follows that $\mathcal M \boxtimes_{\mathcal C} \mathcal M \simeq B\mbox{-}\mathsf{Mod}\mbox{-}A(\mathcal C)$ as $\mathcal C$-bimodule categories. Does this already show that my claim above does not hold?)

Edit: Rephrasing my question, how can the equivalence $\mathcal M \simeq \mathsf{Mod}\mbox{-}A(\mathcal C)$ be made into an equivalence of bimodule categories, if $\mathcal M$ is a bimodule category?

Answer 1

The equivalence is unfortunately not correct: take for example the category $\mathcal{C}=Vec_G$ for some finite group $G$. Take $\mathcal{M}=Vec$ be the category of vector spaces. Then the forgetful functor from $\mathcal{C}$ to $Vec$ enriches $\mathcal{M}$ with the structure of a $\mathcal{C-C}$ bimodule category. If we write $A=KG$, the group algebra, then $\mathcal{M}$ is equivalent to the category of right (left) $A$-modules as a left (right) $\mathcal{C}$-module category. However, the category of $A-A$-bimodules in $\mathcal{C}$ is equivalent to the category $Rep-G$, which is not equivalent to $\mathcal{M}$. One possible approach to the problem is to consider $\mathcal{M}$ as a left module category over the category $\mathcal{D}:=\mathcal{C}\boxtimes\mathcal{C}^{op}$. The category $\mathcal{M}$ is then equivalent to the module category of some algebra in $\mathcal{D}$, which will contain $A\boxtimes B$ as a subalgebra.