This is an exercise of Jacobson's algebra volume II, page $155$:
Let $R$ and $S$ be rings. Let $P$ be a finitely generated projective left $R$-module, $M$ an $R-S$ bimodule, $N$ a left $S$-module. Then there is a group isomorhpism:
$\alpha: hom_{R}(P,M) \otimes_{S} N \rightarrow hom_{R}(P,M \otimes_{S}N)$
such that for $f \in hom_{R}(P,M)$ and $y \in N$ then $\alpha(f \otimes y)$ is the homomorphism $x \mapsto f(x) \otimes {y}$.
I can see why this is a group homomorphism, but why is it a bijection?