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 $((c \stackrel{f}{\to} c'),(d \stackrel{g}{\to} d'))$,
composition of morphisms is defined componentwise by composition in $C$ and $D$.
This operation is the cartesian product in the 1-category Cat.