Definition:Product 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.

The product functor on $\mathbf C$ is the functor $\times: \mathbf C \times \mathbf C \to \mathbf C$ defined by:

where $C \times D$ is a binary product of $C$ and $D$, and $f \times f'$ is the product of $f$ and $f'$.

That it is in fact a functor is shown on Product Functor is Functor.