\begin{definition}[Definition:Pullback Functor]
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:
{{begin-axiom}}
{{axiom|lc= Object functor:
       |m = f^* \alpha := f^* \alpha
       |rc= $f^* \alpha$ is the pullback of $\alpha$ along $f$
}}
{{axiom|lc= Morphism functor:
       |m = f^* \gamma := \gamma'
       |rc= $\gamma': f^* \alpha \to f^* \beta$ is as on Pullback of Commutative Triangle
}}
{{end-axiom}}
Explicitly, $f^* \gamma$ is defined as the unique morphism fitting:
::$\begin{xy}\xymatrix@+1em@L+4px{
 A'
  \ar[rr]^*{f_\alpha}
  \ar[dd]_*{f^* \alpha}
  \ar@{-->}[rd]_*{f^* \gamma}
& &
 A
  \ar[rd]^*{\gamma}
  \ar[dd]^(.4)*{\alpha}
\\ &
 B'
  \ar[ld]^*{f^* \beta}
  \ar[rr] |{\hole} ^(.3)*{f_\beta}
& &
 B
  \ar[ld]^*{\beta}
\\
 C
  \ar[rr]_*{f}
& &
 D
}\end{xy}$
\end{definition}