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.

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.