TITLE: Applying the Yoneda-Lemma to prove the existence of Tensor-products
QUESTION [8 upvotes]: In class the professor said when he came to prove the existence of the tensor-product for $A$-modules ($A$ any ring) that the existence and properties of the tensor-product would be one-liners having proved the Yoneda-Lemma (stated below). He then proceeded to other stuff. 
I wanted to fill in the details of this remark, but I really can't see where to go. Could anyone help me on this? 
Thanks a lot! 
Lemma (Yoneda): Let $D$ be a category, $r$ an object in $D$, and $F : D \rightarrow \mathbf{Set}$ a functor. Then there is a bijection: $$\mathrm{Nat}(\mathrm{Hom}(r, -), F) \simeq F(r)$$
given by $(\alpha: \mathrm{Hom}(r, -) \rightarrow F) \mapsto \alpha_r (1_r)$.

REPLY [7 votes]: As Eric Wofsey says, the Yoneda lemma does not prove that things exist; it proves that things are unique. 
Here is how you can go about proving that things exist using category theory. Let me work with abelian groups for simplicity, although the discussion doesn't really change in general. If $a$ and $b$ are abelian groups, their tensor product $a \otimes b$ is, if exists, the object representing the functor $\text{Bil}(a \times b, -)$ sending an abelian group $c$ to the set of bilinear maps $a \times b \to c$. In general, if $C$ is a cocomplete category, then a functor $F : C \to \text{Set}$ is representable iff it has a left adjoint; if $c$ is the representing object, then the left adjoint takes a set $X$ to the coproduct $\sum_X c$.
Hence what you want is an adjoint functor theorem. A necessary condition for $\text{Bil}(a \times b, -)$ to have a left adjoint is that it preserves limits. This is not hard to check. Next you can apply the presentable adjoint functor theorem: a functor between presentable categories has a left adjoint iff it preserves limits and is accessible; this is a very mild smallness condition and isn't hard to check here either. 
This is quite a bit more annoying than just explicitly constructing the tensor product, but it has the virtue of applying in great generality.