Monoidal 2-structure of bimodule categories

We define a notion of tensor product of bimodule categories and prove that with this product the 2-category of C-bimodule categories for fixed tensor C is a monoidal 2-category in the sense of Kapranov and Voevodsky (1991) [1], . We then provide a monoidal-structure preserving 2-equivalence between the 2-category of C-bimodule categories and Z(C)-module categories (module categories over the center of C). For a finite group G we show that de-equivariantization is equivalent to the tensor product over Rep(G). We derive Rep(G)-module fusion rules and show that the group of invertible Rep(G)-module categories is isomorphic to H2(G,k×), extending results in Etingof et al. [2].