Definition:Composition Functor on Slice Categories

Definition
Let $\mathbf C$ be a metacategory.

Let $C$ and $D$ be objects of $\mathbf C$.

Let $\mathbf C / C$ and $\mathbf C / D$ be the associated slice categories.

Let $g: C \to D$ be a morphism of $\mathbf C$.

Then $g$ defines a composition functor $g_* : \mathbf C / C \to \mathbf C / D$:

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