The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is equivalent to the category of abelian group objects internal to $\mathbf{CRing}_{/R}$, i.e., $\mathbf{Ab}(\mathbf{CRing}_{/R})\simeq \mathbf{Mod(R)}$. We also have that overcategories $\mathbf{Set}_{/A}$ is equivalent to a presheaf category i.e., $\mathbf{Set}_{/A}\simeq \mathbf{Set}^{\mathcal A}$, which we may think of as an action on a set.
My question: Is the equivalence $\mathbf{Ab}(\mathbf{CRing}_{/R})\simeq \mathbf{Mod(R)}$ an enriched analogy of the equivalence $\mathbf{Set}_{/A}\simeq \mathbf{Set}^{\mathcal A}$?