Back to the general level now. Here is the universal property that the posited product object \$$A \times B\$$ must satisfy:

For all other objects \$$C\$$ and with arrows \$$\tau_1: C \rightarrow A\$$ and \$$\tau_2: C \rightarrow B\$$, there exists a _unique_ arrow \$$h: C \rightarrow A \times B\$$ such that \$$\tau_1\$$ and \$$\tau_2\$$ factor through \$$h\$$:

\$\tau_1 = h \triangleright \pi_1\$

\$\tau_2 = h \triangleright \pi_2\$

Note: I am introducing a nonstandard notion here \$$x \triangleright y\$$ to mean \$$x;y\$$ = x-then-y = \$$y \circ x\$$ = y-after-x. The triangle reads nicely to show that the output of morphism x gets fed into the input of morphism y. It's a left-to-right pipeline of morphisms.