Product Functor is Functor

Definition
Let $\mathbf C$ be a metacategory such that any two of its objects admit a product.

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 $\left({C, D}\right)$, we have:

Thus $\times$ preserves identity morphisms.

For composable morphisms $\left({f, f'}\right)$ and $\left({g, g'}\right)$ of $\mathbf C \times \mathbf C$, we have:

Hence $\times$ is a functor.