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 $\gen {f p_1, f' p_2}$.

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Also see

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

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (next): $\S 2.6$