\begin{definition}[Definition:Product of Morphisms]
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 $\left\langle{fp_1, f'p_2}\right\rangle$.
\end{definition}