Isomorphism of Homs How can I show the existence of tensor product mappings. Namely, in Liu there is a problems to show that there exists a unique $A$-linear map
$$f:\operatorname{Hom}_A(M,M^\prime)\otimes_A\operatorname{Hom}_A(N,N^\prime)\to\operatorname{Hom}_A(M\otimes_AN,M^\prime\otimes_AN^\prime)$$
such that $f(u\otimes v)=u\otimes v$.
I guess that here $A$ is a commutative ring with unit. Although it wasn't said on the text.
Is $\operatorname{Hom}(A,B)$ just the set of homomorphism from $A$ to $B$?
Do I need some universal property or should I learn Yoneda's lemma to learn to prove these kind of problems?

Maps out of a tensor product are bilinear maps out of the ordinary product. So the $f$ you want is the linear map corresponding to the bilinear map $\mathrm{Hom}(M,M')\times\mathrm{Hom}(N,N')\to \mathrm{Hom}(M\otimes_A N,M'\otimes_A N')$ which sends $(u,v)$ to $u\otimes v$. The fact that this is bilinear is just the fact that $\otimes$ is an additive functor, which is easy to check by hand: $(au+a'u',v)$ goes to $(au+a'u')\otimes v=a(u\otimes v)+a'(u'\otimes v)$, and similarly in the other argument. (Check it on elements, if you feel the need.)
