Definition:Codomain Functor

Definition
Let $\mathbf C$ be a metacategory.

Let $\mathbf C^\to$ be its morphism category.

The codomain functor is the functor $\mathbf{cod}: \mathbf C^\to \to \mathbf C$ defined by:

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