Identity Morphism of Product

Theorem
Let $\mathbf C$ be a metacategory.

Let $C$ and $D$ be objects of $\mathbf C$, and let $C \times D$ be a product for $C$ and $D$.

Then:


 * $\operatorname{id}_{\left({C \mathop \times D}\right)} = \operatorname{id}_C \times \operatorname{id}_D$

where $\operatorname{id}$ denotes an identity morphism, and $\times$ signifies a product of morphisms.

Proof
By definition of the product morphism $\operatorname{id}_C \times \operatorname{id}_D$, it is the unique morphism making:


 * $\begin{xy}

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

"P";"A" **@{-} ?>*@{>} ?*!/^.8em/{\operatorname{pr}_1}, "P";"A2" **@{-} ?>*@{>} ?*!/_.8em/{\operatorname{pr}_2}, "Q";"B" **@{-} ?>*@{>} ?*!/_.8em/{\operatorname{pr}_1}, "Q";"B2" **@{-} ?>*@{>} ?*!/^.8em/{\operatorname{pr}_2},

"A";"B"  **@{-} ?>*@{>}  ?*!/^.8em/{\operatorname{id}_C}, "A2";"B2" **@{-} ?>*@{>} ?*!/_.8em/{\operatorname{id}_D}, "P";"Q"  **@{--} ?>*@{>} ?*!/_2em/{\operatorname{id}_C \times \operatorname{id}_D}, \end{xy}$

a commutative diagram.

It is immediate by the behaviour of identity morphisms that $\operatorname{id}_{\left({C \mathop \times D}\right)}$ is the unique morphism sought.