I've been dealing with category theory for three weeks now and we just started covering limits and colimits, meanwhile in my geometry class we defined the tensor product of vector spaces.
Then I figured that the tensor product must be a colimit in some category: we have a map $V \times W \to V \otimes W$ which is bilinear, then we have two linear maps $V \to V \otimes W$ and $W \to V \otimes W$, plus the "universal" arrow.
Since I didn't use anything but vector spaces and linear maps, I have that the tensor product is a colimit in $\mathbf{Vect}$.
Did I get anything wrong in my reasoning?
EDIT To clarify: I have a problem with my constructing the two linear maps from $V$ and $W$ to the tensor product (and as per the comments, I now know it's not going to work), but I would like to know how I can see, just by looking at the tensor product, what category it is a colimit of.