# Definition:Product Category

## Definition

Let $\mathbf C$ and $\mathbf D$ be metacategories.

The product category $\mathbf C \times \mathbf D$ is the category with:

 Objects: $\tuple {X, Y}$, for all $X \in \operatorname {ob} \mathbf C$, $Y \in \operatorname {ob} \mathbf D$ Morphisms: $\tuple {f, g}: \tuple {X, Y} \to \tuple {X', Y'}$ for all $f: X \to X'$ in $\mathbf C_1$ and $g: Y \to Y'$ in $\mathbf D_1$ Composition: $\tuple {f, g} \circ \tuple {h, k} := \tuple {f \circ h, g \circ k}$, whenever this is defined Identity morphisms: $\operatorname {id}_{\tuple {X, Y} } := \tuple {\operatorname {id}_X, \operatorname {id}_Y}$

## Also see

• Results about product categories can be found here.