nLab
product category

Definition

For $C$ and $D$ two categories, the product category $C\times D$ is the category whose

• objects are ordered pairs $(c,d)$ with $c$ an object of $C$ and $D$ an object of $D$;

• morphisms are ordered pairs $((c\stackrel{f}{\to }c\prime ),(d\stackrel{g}{\to }d\prime ))$,

• composition of morphisms is defined componentwise by composition in $C$ and $D$.

This operation is the cartesian product in the 1-category Cat.