Composite Functor is Functor

Definition
Let $\mathbf C, \mathbf D$ and $\mathbf E$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be functors.

Let $GF: \mathbf C \to \mathbf E$ be the composition of $G$ with $F$.

Then $GF$ is also 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 $GF$ is shown to be a functor.