Finite semisimple module category over a fusion category

Let $\mathcal{C}$ be a fusion category over $k$. A finite semisimple right $\mathcal{C}$-module category consists of a $k$-linear abelian category $\mathcal{M}$, semisimple in the sense of Semisimple objects and semisimple abelian categories, with finitely many isomorphism classes of simple objects and finite-dimensional $k$-vector spaces ${\mathrm{Hom}}_{\mathcal{M}}(m,n)$ for all $m,n\in \mathcal{M}$, together with a bilinear action functor $\odot :\mathcal{M}\times \mathcal{C}\to \mathcal{M}$ and natural isomorphisms ${\alpha }_{m,X,Y}:(m\odot X)\odot Y⟶m\odot (X\otimes Y),{\rho }_m:m\odot \mathbf{1}⟶m,$ which satisfy the module pentagon and triangle equations obtained from the associator and unitors of $\mathcal{C}$. The left-action convention is defined by reversing the order of the action factors.

A module functor $F:\mathcal{M}\to \mathcal{N}$ is a linear functor with natural action comparisons $F(m\odot X)\cong F(m)\odot X$ satisfying the module associativity and unit equations. A module-natural transformation is a natural transformation commuting with those comparisons. These objects, functors, and transformations form a 2-category denoted $\mathrm{Mod}(\mathcal{C})$ with ordinary composition of module functors and horizontal and vertical composition of module-natural transformations. The action comparison for a composite is the composite of the two action comparisons; identity functors have identity comparisons.

This definition alone does not assert that there are finitely many equivalence classes of simple module categories, nor that $\mathrm{Mod}(\mathcal{C})$ is a fusion 2-category.