I've been given an arbitrary extension $S \subset R$ of associative algebras with units, and I need to construct a right adjoint functor to the restriction functor $F = \text{res}_S^R ∶ R\mathcal{-Mod} \rightarrow S\mathcal{-Mod}$
I think the functor $G=\text{Hom}(R,-):S\mathcal{-Mod} \rightarrow R\mathcal{-Mod}$ is right adjoint to $F$. But I have no idea how to show that the following is a natural isomorphism (i.e. I'm having trouble defining the map) $$\phi:\text{Hom}_{R\mathcal{-Mod}}(X,\text{Hom}(R,Y))\rightarrow \text{Hom}_{S\mathcal{-Mod}}(\text{res}_S^R(X),Y)$$