  In [arXiv:1509.02937], the notion of a module tensor category was introduced
as a braided monoidal central functor $F\colon \mathcal{V}\longrightarrow
\mathcal{T}$ from a braided monoidal category $\mathcal{V}$ to a monoidal
category $\mathcal{T}$, which is a monoidal functor $F\colon
\mathcal{V}\longrightarrow\mathcal{T}$ together with a braided monoidal lift
$F^Z\colon \mathcal{V}\longrightarrow Z(\mathcal{T})$ to the Drinfeld center of
$\mathcal{T}$. This is a categorification of a unital associative algebra $A$
over a commutative ring $R$ via a ring homomorphism $f\colon R\longrightarrow
Z(A)$ 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
[arXiv:1509.02937] by explicitly writing down an equivalence of 2-categories.
