In wikipedia the definition of tensor product by the universal property can be found here. It says that:
The tensor product of two vector spaces V and W is a vector space denoted as V ⊗ W, together with a bilinear map ⊗: (v,w) ↦ v ⊗ w from V × W to V ⊗ W , such that, for every bilinear map h: V × W → Z, there is a unique linear map h': V ⊗ W → Z such that h = h' ∘ ⊗ (that is, h( v , w ) = h'( v ⊗ w ) for every v ∈ V and w ∈ W.
In the same section of this definition, they say that it is a example of construction by the universal property of category theory, that can be found here. But it seems to me the definition of tensor product is close but not completely alike the definition by universal properties. In the definition of tensor product they don't explicitly say what are the categories and functors involved and I got a little confused when I tried to understand what they are.
If I would have to guess, I think all the construction above is happening in the category of vector spaces and multilinear functions, and there is a forgetting $U$ and free functor $F$ from it to the category $Set$, so that $V \otimes W = FU(V \times W)$.
But, I think my guess is wrong. Can someone help?
Universal properties can always be phrased in terms of the question of whether some functor or functors is representable. Here the question that tensor products answer is whether the functor sending a vector space $Z$ to the set $\text{Bilin}(V, W; Z)$ of bilinear maps $V \times W \to Z$ is representable, and the answer is yes, by the tensor product $V \otimes W$.