Product Functor is Functor

Definition
Let $\mathbf C$ be a metacategory with binary products.

Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.

Let $\times: \mathbf C \times \mathbf C \to \mathbf C$ be the product functor.

Then $\times$ is a functor.

Proof
For any pair of objects $\tuple {C, D}$:

Thus $\times$ preserves identity morphisms.

For composable morphisms $\tuple {f, f'}$ and $\tuple {g, g'}$ of $\mathbf C \times \mathbf C$:

Hence $\times$ is a functor.