Product Category is Product in Category of Categories

Theorem
Let $\mathbf{Cat}$ be the category of categories.

Let $\mathbf C$ and $\mathbf D$ be small categories, and let $\mathbf C \times \mathbf D$ be their product category.

Then $\mathbf C \times \mathbf D$ is a product of $\mathbf C$ and $\mathbf D$ in $\mathbf{Cat}$.