Do functors induce morphisms between internal homomorphisms?

Question

Consider a closed symmetric monoidal abelian category $A$, a monoid $R$ in this category, and a functor $F\colon\text{Mod}(R)\to A$. Then there is, for all $R$-module objects $M$ and $N$, a map $\text{Hom}_{R}(M,N)\to\text{Hom}(F(M),F(N))$. Is it possible to construct a similar morphism $\underline{\text{Hom}}_{R}(M,N)\to\underline{\text{Hom}}(F(M),F(N))$ between the internal homomorphisms?