Definition:Product Category

Definition
Let $\mathcal C$ and $\mathcal D$ be categories.

Let $\operatorname{Hom}$ denotes the hom classes of a category.

Define the product category $\mathcal C \times \mathcal D$ to be the category with:
 * Objects $(X,Y)$, where $X \in \operatorname{ob}\mathcal C$, $Y \in \operatorname{ob}\mathcal D$
 * Morphisms $(f,g) \in \operatorname{Hom}((X,Y),(X',Y'))$ where $f \in \operatorname{Hom}(X,X')$, $g \in \operatorname{Hom}(Y,Y')$
 * Composition given by $(f,g) \circ (h,k) = (f \circ h, g \circ k)$ whenever this is defined
 * Identity morphisms $\operatorname{id}_{(X,Y)} = (\operatorname{id}_X,\operatorname{id}_Y)$

By Product Category is a Category, this is indeed a category.