Understanding Product Maps $f \times g : A\times B \to C\times D$

One way to understand these maps is to think of them as tuples of set maps $f: A \to C, g : B \to D$.  So if you're having trouble despite, the following categorical explanation, just think of them that way - the way from which the categorical construction is derived.

Suppose that the products $A\times B$ and $C\times D$ exist in our category.  This means that by definition:



Or in English: $A \xleftarrow{p_1} A \times B \xrightarrow{p_2} B$ is a product diagram if and only if for every glued in diagram $A \xleftarrow{x_1} X \xrightarrow{x_2} B$, there exists a unique map (i.e. UMP property here) $u: X \to A\times B$ such that everything commutes, i.e. $p_i u = x_i$ for each $i=1,2$.

Now, we simply glue in the arrows $f$ and $g$:


as well as the product $C \times D$ and its projection maps:


Can you finish the derivation from here?  If not, then please continue.

Now, instead of the UMP for $A\times B$, use the UMP of $C\times D$ to get that there exists a unique map $u: A \times B \to C\times D$ such that everything commutes, as in:


Now we simply define the notation $f \times g$ to mean that very map $u$. And that's how product maps are derived. Note that $f\circ p_i = q_i \circ u$ or in terms of set maps and tuples we have that $f\times g (x,y) = (f(x), g(y))$.

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