\begin{definition}[Definition:Product Category]
Let $\mathbf C$ and $\mathbf D$ be metacategories.
The '''product category''' $\mathbf C \times \mathbf D$ is the category with:
{{DefineCategory
| ob = $\tuple {X, Y}$, for all $X \in \operatorname {ob} \mathbf C$, $Y \in \operatorname {ob} \mathbf D$
| mor = $\tuple {f, g}: \tuple {X, Y} \to \tuple {X', Y'}$ for all $f: X \to X'$ in $\mathbf C_1$ and $g: Y \to Y'$ in  $\mathbf D_1$
| comp = $\tuple {f, g} \circ \tuple {h, k} := \tuple {f \circ h, g \circ k}$, whenever this is defined
| id = $\operatorname {id}_{\tuple {X, Y} } := \tuple {\operatorname {id}_X, \operatorname {id}_Y}$
}}
\end{definition}