Let $\mathcal{C}$ be a monoidal category and $M$ a left module category over $\mathcal{C}$. That is, a category equipped with an exact bifunctor $F:\mathcal{C}\otimes M\rightarrow M$ satisfying some conditions (see Ostrik's paper at <https://arxiv.org/abs/math/0111139> for a nice exposition). My question is: What conditions on $\mathcal{C}$ ensure that $M$ is monoidal?



When is a Module category monoidal?
Ostrik's paper shows that every indecomposable module category comes from a connected algebra $A$ given by the internal action endomorphisms. If $A$ can be chosen to be commutative in the center $Z(\mathcal C$), then the module category has the structure of a monoidal category, see e.g. here <http://arxiv.org/abs/1006.0569> and references therein.


If $\mathcal C$ is braided it is enough to ask to ask $A$ to be commutative in $\mathcal C$. There are actually two possible ways to do this.


