TITLE: Universal property for tensor product in an arbitrary category
QUESTION [12 upvotes]: The tensor product for vector spaces is defined by a universal property (diagram from Wikipedia)

for every bilinear map $h$ there exists a unique linear map $\tilde h$ such that the diagram commutes. ($\varphi$ is part of the definition of the tensor product.) 
This is kind of a funny diagram, because bilinear maps aren't linear maps, so it's not a diagram in Vect. Because of this, it doesn't seem obvious how to transfer the concept to an arbitrary category.
My question is, can this be done? That is, can the notion of bilinear map itself be defined in category-theoretic terms, starting from the objects and morphisms of Vect? Or else can the universal property for the tensor product be expressed without using a notion like "bilinear map" that's external to the category Vect?
In other words, can the tensor product be defined in such a way that, given an arbitrary category, it becomes a well-defined question whether it has tensor products or not, and if so what they are.
(Note: the monoidal operator in a monoidal category is sometimes called a tensor product, but this is a somewhat different thing, because in general there are many choices of monoidal product. For this question I'm interested in whether the definition given above can generalise in such a way that it's uniquely defined for any category, if it exists.)

REPLY [3 votes]: 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.