Definition:Slice Functor

Definition
Let $\mathbf C$ be a metacategory.

Let $\mathbf{Cat}$ be the category of categories.

The slice functor is the functor $\mathbf C / \cdot: \mathbf C \to \mathbf{Cat}$ defined by:

where $\mathbf C / C$ is a slice category, and $f_*$ is the composition functor defined by $f$.

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