Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true

- ($R$-Mod) $\simeq$ ($S$-Mod).
- $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective generator in ($R$-Mod).

A tensor category is a categorical analogue of a ring. Two tensor categories $C$ and $D$ are said to be categorical Morita equivalent if there is an exact $C$-module category $M$ and a tensor equivalence [1, Definition 7.12.17]

$$ D^{op} \simeq C^\star_M. $$

This definition resembles the second condition in the classical case. Thus my **question**:

In this case, do $C$ and $D$ have equivalent categories of module categories?

**Reference**

- [1] Tensor Categories-[Etingof, Gelaki, Nikshych, and Ostrik]