Tensor functors between Morita duals of fusion categories

Understand

Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual fusion category of $\mathcal{C}$ with respect to $\mathcal{M}$.

• We describe tensor functors between two arbitrary duals $\mathcal{C}^*_\mathcal{M}$ and $\mathcal{D}^*_\mathcal{N}$ in terms of data associated to $\mathcal{C}$ and $\mathcal{D}$.
• We apply the results to $G$-equivariantizations of fusion categories and group-theoretical fusion categories.
• We describe the orbits of the action of the Brauer-Picard group on the set of module categories and we propose a categorification of the Rosenberg-Zelinsky sequence for fusion categories.