Universal property for tensor product in an arbitrary category

In general, there isn't a unique tensor (monoidal product) on a category. For example, any category with binary products and coproducts has the structure of a monoidal category coming from each.

There are two (fairly) simple ways to force the tensor to be unique, however.

First, a method that doesn't really introduce anything you haven't seen, but maybe just punts the question. You may be familiar with the idea that the set of linear transformations from $A$ to $B$ can itself be given the structure of a vector space (call it $[A, B]$). Given such an internal hom (which is functorial in its arguments), one can define the tensor of two objects to be an object $A \otimes B$ such that $\hom(A \otimes B, C) \cong \hom(A, [B, C])$. Phrasing this a universal property, we should have a morphism $\varphi: A \to [B, A \otimes B]$ such that for any morphism $f: A \to [B, C]$, there exists a unique $g: A \otimes B \to C$ such that $[B, g] \circ \varphi = f$.

It turns out that for technical reasons, for this to have good properties (like associativity), this needs to be upgraded to an isomorphism $[A, [B, C]] \cong [A \otimes B, C]$, but for the case of vector spaces, that's not much harder to prove.

So what does this have to do with multilinear maps? It turns out a multilinear map $A \times B \to C$ is the same as a linear map $A \to [B, C]$, so saying that these in turn correspond to linear maps $A \otimes B \to C$ is simply expressing the universal property above.

A more principled way to do this requires that we generalize categories to multicategories. A multicategory is like a category, but now our domains are finite lists of objects. That is, a morphism can go from the list $(A_1, A_2, ..., A_n)$ to an object $B$. For the case of vector spaces, we can define the maps $(A_1, A_2, ..., A_n) \to B$ to be the multilinear maps $A_1 \times A_2 \times ... \times A_n \to B$. (Note that in the special case where $n = 0$, this is simply an element of $B$, or more precisely, a function from a singleton set to $B$ with no linearity requirements).

Then the tensor product on this multicategory, if it exists, is an object $A \otimes B$ together with a (mutli)map $\varphi : (A, B) \to A \otimes B$ such that for any map $f : (A, B) \to C$, there is a unique map $g : A \otimes B \to C$ such that $g \circ \varphi = f$. Put another way, there should be a natural isomorphism $\hom((A, B), C) \cong \hom(A \otimes B, C)$.

The properties of multicategories (see the link above) ensure that this tensor is well-behaved, including associativity. If you introduce an empty tensor (an object $I$ such that $\hom((), C) \cong \hom(I, C)$), this empty tensor behaves as a unit for the tensor product ($A \otimes I \cong I \otimes A \cong A$).

We can make this (and similar) diagrams rigorously live in a category, namely the one that connects up $Vect\times Vect$ with $Vect$ by bilinear maps $U\times V\to W$ as additional morphisms $(U,V)\to W$, and define their compositions in a straightforward way.

Observe that the tensor product $U\otimes V$ is given as the reflection of $(U,V)$ in $Vect$.

This construction, to put 'heteromorphisms' in one direction in between (the disjoint union of) two categories is called (the 'collage' of) a profunctor.

The Yoneda lemma governs this realms. Recall that it says that, for a functor $F:C\to Set$ and an object $x\in C$, there is a natural bijection $$ \Phi:\text{Nat}(\hom(-,x),F)\xrightarrow{\sim}Fx. $$ What you've figured is that a tensor product of $V$ and $W$ can be defined as a representation for the functor $\text{Bilin}(V,W;-):Vect\to Set$ which takes a vector space $U$ and spits the set of bilinear maps $V\times W\to U$: $$ \text{Bilin}(V,W;-) \cong Vect(V\otimes W,-) $$ The Yoneda lemma then says that each such natural isomorphism comes from an element of $\text{Bilin}(V,W;V\otimes W)$, which is a bilinear map $\otimes:V\times W\to V\otimes W$. This is the usual projection on the tensor product.

Moreover, the proof of the Yoneda lemma says that the following diagram commutes: $$ \begin{array}{ccc} Vect(V\otimes W,V\otimes W) & \xrightarrow{} & Bilin(V,W,V\otimes W) \\ \downarrow & & \downarrow \\ Vect(V\otimes W,U)&\xrightarrow{}&Bilin(V,W,U) \end{array} $$ Horizontally we use the natural isomorphism from the Yoneda lemma, and vertically, composition with any linear transformation $f:V\otimes W\to U$.

Starting with the identity $id:V\otimes W\to V\otimes W$, commutativity of this diagrams witness precisely the universal property of the tensor product, with uniqueness coming from the horizontal maps being isomorphisms: $$ \Phi(f) = f\circ \otimes $$

The bilinear map $\bar{f}$ is $\Phi(f)$. Credits to Emily Riehl for explaining this stuff in Category Theory in Context.