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Let $V$ and $W$ be linear spaces. According to Wikipedia, the universal definition of the tensor product $V \otimes W$ satisfies the following property:

There is a bilinear map (i.e., linear in each variable $v$ and $w$) $\varphi : V\times W\to V \otimes W$ such that given any other vector space $Z$ together with a bilinear map $h:V\times W\to Z$, there is a unique linear map $\tilde{h}:V\otimes W\to Z$ verifying $h=\tilde{h}\circ \varphi$.

I am intrigued by this definition. The Cartesian product is supposed to have a categorial definition, which I have not comprehended so far, but the I have not seen a categorial definition of bilinearity. In this sense, the proposed categorial definition of tensor products seems to cheat.

Can somebody work out the tensor product in purely categorial terms?

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No, since this would mean that we have a notion of tensor product in every category (different from the cartesian product), which is not the case. What is the tensor product of two homotopy types? The notion of bilinearity really depends on the specific structure of our category of vector spaces. That being said, there is a quite general notion of bihomomorphisms in concrete categories and tensor products classifying them:

B. Banaschewski and E. Nelson, Tensor products and bimorphisms, Canad. Math. Bull. 19 (1976) 385-401.

Notice that a bilinear map is not a map $V \times W \to Z$ of vector spaces, but rather a map $|V| \times |W| \to |Z|$ between the underlying sets, with additional properties. This explains the concreteness assumption.

There is also a notion of bihomomorphisms in monoidal categories. This has been studied by Anders Kock, Gavin Seal and others (see for example arXiv/1205.0101).

By the way, you should really understand cartesian products before tensor products or reading any of the mentioned papers.

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$\require{AMScd}$ I think your trouble comes from the fact that you are trying to see the diagram $$\begin{CD} V \times W @>>> V\otimes W \\ @| @VVV \\ V \times W @>>> Z \end{CD}$$ where horizontal maps are bilinear maps, inside the category of vector spaces over $k$. What you cannot.

What the universal property says is that the functor $$ \mathrm{Bil}\, (V, W ; - ) \colon \mathbf{Vect}_k \to \mathbf{Set},\, Z \mapsto \{ k\text{-bilinear map } V\times W \to Z\} $$ is representable (by an object that we denote $V \otimes W$).

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