Definition:Pullback Functor

Definition
Let $\mathbf C$ be a metacategory having all pullbacks.

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

Let $\mathbf C \mathop / C$ and $\mathbf C \mathop / D$ be the slice categories over $C$ and $D$, respectively.

The pullback functor $f^* : \mathbf C \mathop / D \to \mathbf C \mathop / C$ associated to $f$ is defined by:

That $f^*$ is a functor is shown on Pullback Functor is Functor.