Module monoidal categories as categorification of associative algebras

In [12], the notion of a module tensor category was introduced as a braided monoidal central functor $F:V⟶T$ from a braided monoidal category $V$ to a monoidal category $T$, which is a monoidal functor $F:V⟶T$ together with a braided monoidal lift $F^Z:V⟶Z\left(T\right)$ to the Drinfeld center of $T$. This is a categorification of a unital associative algebra A over a commutative ring R via a ring homomorphism $f:R⟶Z\left(A\right)$ into the center of A. In this paper, we want to categorify the characterization of an associative algebra as a (not necessarily unital) ring A together with an R–module structure over a commutative ring R, such that multiplication in A and action of R on A are compatible. In doing so, we introduce the more general notion of non–unital module monoidal categories and obtain 2–categories of non–unital and unital module monoidal categories, their functors and natural transformations. We will show that in the unital case the latter definition is equivalent to the definition in [12] by explicitly writing down an equivalence of 2–categories.