The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here $\otimes$ denotes the component-wise tensor product of simplicial vector spaces (it gives to the category $sVect$ of simplicial vector spaces a symmetric monoidal structure).

But there is a natural structure of simplicial algebra on $sHom(V,V)$ (the internal $Hom$ induced by the product $\otimes$). So we can also define structure of a simplicial module over $A$ on $V$ by fixing a morphism of simplicial algebras $A\to sHom(V,V)$.

I think we can do it in more generality, for any symmetric monoidal category $C$ with internal $Hom$. If we have a monoid $A$ in $C$ and some object $V\in C$, then we can define the structure of $A$-module on $V$ in two different ways: via $A\otimes V\to V$ and $A\to \underline{Hom}(V,V)$, where $\underline{Hom}$ is the internal $Hom$.

The

questionis: are these two definitions equivalent for any symmetric monoidal category with internal $\underline{Hom}$? If not, is it true for simplicial vector spaces?

Thank you very much!