TITLE: Adjointness of Hom and Tensor
QUESTION [14 upvotes]: Could someone provide me a link to the proof of the adjointness of Hom and Tensor. I did an extensive google search but could not find anything self contained that presented the proof in full generality (or at least the generality I know). 
Let $R\to S$ be a ring homomorphism, let $M,N$ be $S$-modules and $Q$ an $R$-module. Then, we have 
$$\textrm{Hom}_R(M\otimes_S N,Q) \cong \textrm{Hom}_S(M,\textrm{Hom}_R(N,Q)$$

REPLY [13 votes]: Let $f \in \operatorname{Hom}_R(M\otimes_S N,Q)$.  We define $g \in \operatorname{Hom}_S(M, \operatorname{Hom}_R(N,Q))$ by:
$$g(m)(n)=f(m \otimes n)$$
Similarly, if $g$ is defined, we can easily define $f$.
I'll leave it to you to prove that this map between $f$ and $g$ actually goes to the appropriate sets, but this is the basic argument. As mentioned by one of the comments, it does depend on how you define the tensor product.