  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.
