Identity Functor is Functor

Theorem
Let $\mathbf C$ be a metacategory.

Let $\operatorname{id}_{\mathbf C}: \mathbf C \to \mathbf C$ be the identity functor on $\mathbf C$.

Then $\operatorname{id}_{\mathbf C}$ is a functor.

Proof
Let $f, g$ be morphisms of $\mathbf C$ such that $g \circ f$ is defined.

Then:

Also, for any object $C$ of $\mathbf C$:

Hence $\operatorname{id}_{\mathbf C}$ is shown to be a functor.