I've seen two introductions to the tensor product - in one, tensor products are maps to a field:
$V^\ast\otimes V^\ast:V\times V\rightarrow F$ because this is $({v^\ast}' w')(v^\ast w)$ which is the product of two real numbers/elements of the field
In the universal property, tensor products are being mapped (by a linear function $\hat{f}$, and $f$is multilinear):
$f:A\times B\to C$
$\otimes:A\times B\to A\otimes B$
$\hat{f}:A\otimes B\to C$
An image is given in the article "How to lose your fear of tensor products" for the second, and the first approach from XylyXylyX on youtube "What is a tensor? Lesson 5". How are these related, and why? Is the first the only way a tensor product can act, and if so how is that derived from the universal property?
Any insight is welcome!