Definition:Product of Morphisms

Definition
Let $\mathbf C$ be a metacategory.

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


 * $\begin{xy}\xymatrix@R-1em@C+1em@L+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@+1em@L+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 $\gen {f p_1, f' p_2}$.

Also see

 * Definition:Binary Product (Category Theory)
 * Definition:Product Functor