Definition:Product of Morphisms

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Let $\mathbf C$ be a metacategory.

Let $A, A'$ and $B, B'$ be pairs of objects admitting binary products:

$\begin{xy}\[email protected]@[email protected]+3px{ A & A \times A' \ar[l]_*+{p_1} \ar[r]^*+{p_2} & A' \\ B & B \times B' \ar[l]_*+{q_1} \ar[r]^*+{q_2} & B' }\end{xy}$

Let $f: A \to B$ and $f': A' \to B'$ be morphisms.

The product morphism of $f$ and $f'$, denoted $f \times f'$, is the unique morphism making the following diagram commute:

$\begin{xy}\xymatrix@[email protected]+3px{ A \ar[d]_*+{f} & A \times A' \ar[l]_*+{p_1} \ar[r]^*+{p_2} \ar@{-->}[d]^*+{\hskip{1.3em} f \times f'} & A' \ar[d]^*+{f'} \\ B & B \times B' \ar[l]^*+{q_1} \ar[r]_*+{q_2} & B' }\end{xy}$

Thus we see that $f \times f'$ is the morphism $\left\langle{fp_1, f'p_2}\right\rangle$.

Also see