Product Category is Category

Theorem
Let $\mathbf C$ and $\mathbf D$ be metacategories.

Then the product category $\mathbf C \times \mathbf D$ is a metacategory.

Proof
Let $\left({X,Y}\right), \left({X',Y'}\right) \in \mathbf C \times \mathbf D$.

Let $\left({f,g}\right) : \left({X,Y}\right) \to \left({X',Y'}\right)$ and $\left({h,k}\right) : \left({X',Y'}\right) \to \left({X,Y}\right)$ be morphisms.

Let $\operatorname{id}_X$, $\operatorname{id}_Y$ be the identity morphisms for the objects $X$ and $Y$ respectively.

Then:

Similarly:

Therefore, $\left({\operatorname{id}_X, \operatorname{id}_Y}\right)$ satisfies the property of an identity morphism.

Let $\left({f, g}\right)$, $\left({h, k}\right)$ and $\left({\ell, m}\right)$ be composable morphisms of $\mathbf C \times \mathbf D$. We have:

Therefore, composition of morphisms in $\mathbf C \times \mathbf D$ is also associative.