Let $R$ be a CRing and let $M,N$ be $R$-modules. Let $M^*:=Hom_R(M,R)$. I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces:
$M^*\otimes_R N\cong Hom_R(M,N)$.
I can give a proof of this using a basis, but in what generality can we say this kind of thing? Does it have a nice arrow-theoretic proof?