Product category

Remarks

The category axioms hold because they hold in each coordinate. There are projection functors $\mathcal{C} \times \mathcal{D} \to \mathcal{C}$ and $\mathcal{C} \times \mathcal{D} \to \mathcal{D}$. Example: a monoid $M$ is a category with one object; the product of the one-object categories of $M$ and $N$ is the one-object category of the product monoid $M \times N$. Indeed its only object is the pair of the two objects, and its morphisms are the pairs $(m, n)$ multiplied coordinatewise. The set of objects of $\mathcal{C} \times \mathcal{D}$ is the Cartesian product of the sets of objects when both categories are small. A morphism $(f, g)$ is an isomorphism exactly when $f$ and $g$ are both isomorphisms.