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 binary 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}\xymatrix@+1em@L+5px{

C \ar[d]_*+{\operatorname{id}_C} & C \times D \ar[l]_*+{\operatorname{pr}_1} \ar[r]^*+{\operatorname{pr}_2} \ar@{-->}[d]^*+{\hskip{2.2em} \operatorname{id}_C \times \operatorname{id}_D} & D \ar[d]^*+{\operatorname{id}_D}

\\ C & C \times D \ar[l]^*+{\operatorname{pr}_1} \ar[r]_*+{\operatorname{pr}_2} & 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.