# $A$-$B$-bimodules as $A \otimes B^{op}$-modules in a tensor category

In the book "Tensor Categories" by Etingof, Gelaki, Nikshych and Ostrik, the following is stated (Remark 7.8.26): "Note that in the categorical setting, we cannot, in general, define ($A,B$)-bimodules as modules over $A\otimes B^{op}$, since neither the opposite algebra nor the tensor product of algebras is defined in a general multitensor category."

Now my question is, what are the minimal requirements on a tensor category $\mathcal C$ to ensure an equivalence $A\mbox{-}\mathsf{Mod}\mbox{-}B(\mathcal C) \simeq (A\otimes B^{op})\mbox{-}\mathsf{Mod}(\mathcal C)$ of module categories, i.e. when are opposite algebras and the tensor product of algebras defined?

• To take opposite algebras you need a braided monoidal structure. I think this is also enough to define an algebra structure on the tensor product of algebras. – Qiaochu Yuan Mar 7 '17 at 19:16
• See remark: 4.2 (12 in arXiv version) in V. Ostrik, Module categories, weak Hopf algebras and modular invariants, math.QA/0111139. – Marcel Bischoff Mar 8 '17 at 2:10