I am just getting to know some category theory.
My understanding of the "universal property" (based on this Wikipedia article) is that it is characterized in terms of an "initial morphism" or "terminal morphism". As an example, suppose $\mathcal{B}\in \text{Set}$, $\text{Vect}$ the category of vector spaces over some field, $\mathcal{F}:\text{Vect} \to \text{Set}$ the forgetful functor, and $V_{\mathcal{B}}$ a complex vector space with basis $\mathcal{B}$. Then the fact that a vector space is free over its basis implies that given $W\in \text{Vect}$, there is a unique $\tau \in \text{Hom(Vect)}$ such that the following diagram commutes:
$$\begin{array}{c} \mathcal{B} & \overset{g}{\longrightarrow} & \mathcal{F}(V_{\mathcal{B}})& V_{\mathcal{B}}\\ & \searrow_{f} & \downarrow_{\mathcal{F}(\tau)} & \downarrow{\tau} \\ &&\mathcal{F}(W) & W \end{array}$$
Similarly, we can define the direct sum of vector spaces by the below diagram, where $\Delta: \text{Vect} \to \text{Vect}\times \text{Vect}$ is the diagonal map:
$$\begin{array}{c} U,V & \overset{g_1, g_2}{\longrightarrow} & \Delta(U\oplus V )& U \oplus V\\ & \searrow_{f_1, f_2} & \downarrow_{\Delta(\tau)} & \downarrow{\tau} \\ &&\Delta(W) & W \end{array}$$
In learning about the tensor product recently, I learned that it is defined via a universal property with respect to bilinear maps. But I am not able to nail down what the relevant functor is. In a diagram like:
$$\begin{array}{c} U\times V & \overset{\iota}{\longrightarrow} & \mathcal{F}(U\otimes V )& U \otimes V\\ & \searrow_{f} & \downarrow_{\mathcal{F}(\tau)} & \downarrow{\tau} \\ &&\mathcal{F}(W) & W \end{array}$$
$f$ and $\iota$ should be bilinear maps somehow, but what category does $U\times V$ belong to, and what is the functor $\mathcal{F}$?
