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:

			Object functor:		$\ds f^* \alpha := f^* \alpha $				$f^* \alpha$ is the pullback of $\alpha$ along $f$
			Morphism functor:		$\ds f^* \gamma := \gamma' $				$\gamma': f^* \alpha \to f^* \beta$ is as on Pullback of Commutative Triangle

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}

& &

}\end{xy}$