# Categorical Morita equivalence implies equivalence of module categories?

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

1. ($$R$$-Mod) $$\simeq$$ ($$S$$-Mod).
2. $$S \simeq Hom_R(M,M)$$, 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]
• There’s some technical issues you need to be careful about. Firstly, the collection of module categories forms a 2-category! Second you need to be careful about how you build that 2-category (probably you want the right exact functors). – Noah Snyder Jan 22 at 16:00

Theorem 7.12.16. Let $$M$$ be a faithful exact module category over $$C$$. The $$2$$-functor
$$(7.36) \quad\quad N \mapsto Fun_{C}(M,N): Mod(C) \to Mod((C_M^\star)^{op})$$
is a $$2$$-equivalence.