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:

Explicitly, $f^* \gamma$ is defined as the unique morphism fitting:


 * $\begin{xy}\xymatrix@+1em@L+4px{

A' \ar[rr]^*{f_\alpha} \ar[dd]_*{f^* \alpha} \ar@{-->}[rd]_*{f^* \gamma} & & A \ar[rd]^*{\gamma} \ar[dd]^(.4)*{\alpha}

\\ & B' \ar[ld]^*{f^* \beta} \ar[rr] |{\hole} ^(.3)*{f_\beta} & & B \ar[ld]^*{\beta}

\\ C \ar[rr]_*{f} & & D }\end{xy}$

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