Definition:Product of Morphisms

Definition
Let $\mathbf C$ be a metacategory.

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


 * $\begin{xy}

<-4em,0em>*+{A}         = "A", <0em,0em>*+{A \times A'} = "P", <4em,0em>*+{A'}         = "A2",

"P";"A" **@{-} ?>*@{>} ?*!/^.8em/{\scriptstyle p_1}, "P";"A2" **@{-} ?>*@{>} ?*!/_.8em/{\scriptstyle p_2},

<-4em,-2em>*+{B}         = "B", <0em,-2em>*+{B \times B'} = "Q", <4em,-2em>*+{B'}         = "B2",

"Q";"B" **@{-} ?>*@{>} ?*!/^.8em/{\scriptstyle q_1}, "Q";"B2" **@{-} ?>*@{>} ?*!/_.8em/{\scriptstyle q_2}, \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}

<-5em,0em>*+{A}         = "A", <0em,0em>*+{A \times A'} = "P", <5em,0em>*+{A'}         = "A2", <-5em,-5em>*+{B}         = "B", <0em,-5em>*+{B \times B'} = "Q", <5em,-5em>*+{B'}         = "B2",

"P";"A" **@{-} ?>*@{>} ?*!/^.8em/{p_1}, "P";"A2" **@{-} ?>*@{>} ?*!/_.8em/{p_2}, "Q";"B" **@{-} ?>*@{>} ?*!/_.8em/{q_1}, "Q";"B2" **@{-} ?>*@{>} ?*!/^.8em/{q_2},

"A";"B"  **@{-} ?>*@{>}  ?*!/^.8em/{f}, "A2";"B2" **@{-} ?>*@{>} ?*!/_.8em/{f'}, "P";"Q"  **@{--} ?>*@{>} ?*!/_1.4em/{f \times f'}, \end{xy}$

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

Also see

 * Product (Category Theory)
 * Product Functor