Pullback Functor is Functor

Theorem
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.

Let $f^* : \mathbf C \mathop / D \to \mathbf C \mathop / C$ be the pullback functor defined by $f$.

Then $f^*$ is a functor.