Let $R$ be a ring with identity, $M$ and $M'$ two right $R$-module, $N$ and $N'$ two left $R$-module. There is a natural way to define a homomorphism
$$f:\hom_R(M,M') \otimes \hom_R(N,N')\to \hom(M\otimes_R N, M'\otimes_RN').$$
My question is that, is $f$ always monic, epic, or isomorphic? And if any answer is no, then is there any characterization of the case when $f$ is so?
Since you are considering not necessarily commutative ring and thus is forced to taking hom-set and tensor product of abelian group, it's not reasonable to expect that $\hom_R(M,M')\otimes\hom_R(N,N')$ and $\hom(M\otimes_RN,M'\otimes_RN')$ are comparable. For example, taking $M:=R_R,N:=_RR$, then the two become $R\otimes_{\mathbb Z}\hom_R(M',N')$ and $\hom_{\mathbb Z}(R,M'\otimes_RN')$, which can be very different.
I suggest that you could consider a similar and more natural question: for commutative ring $R$, when $f:\hom_R(M,M')\otimes_R\hom_R(N,N')\to\hom_R(M\otimes_RN,M'\otimes_RN')$ is epic, monic, or isomorphic, of which a link of the partial solution has been given in the comment.