\begin{definition}[Definition:Pullback (Category Theory)]
Let $\mathbf C$ be a metacategory.
Let $f: A \to C$ and $g: B \to C$ be morphisms with common codomain.
A '''pullback''' of $f$ and $g$ is a commutative diagram:
::$\begin{xy}\xymatrix{
 P
  \ar[r]^*+{p_1}
  \ar[d]_*+{p_2}
&
 A
  \ar[d]^*+{f}
\\
 B
  \ar[r]_*+{g}
&
 C
}\end{xy}$
such that $f \circ p_1 = g \circ p_2$, subject to the following UMP:
:For any commutative diagram:
:::$\begin{xy}\xymatrix{
 Q
  \ar[r]^*+{q_1}
  \ar[d]_*+{q_2}
&
 A
  \ar[d]^*+{f}
\\
 B
  \ar[r]_*+{g}
&
 C
}\end{xy}$
:there is a unique morphism $u: Q \to P$ making the following diagram commute:
:::$\begin{xy}\xymatrix@+1em{
 Q
  \ar@/^/[rrd]^*+{q_1}
  \ar@/_/[ddr]_*+{q_2}
  \ar@{-->}[rd]^*+{u}
\\
&
 P
  \ar[r]_*+{p_1}
  \ar[d]^*+{p_2}
&
 A
  \ar[d]^*+{f}
\\
&
 B
  \ar[r]_*+{g}
&
 C
}\end{xy}$
In this situation, $p_1$ is called the '''pullback of $f$ along $g$''' and may be denoted as $g^* f$.
Similarly, $p_2$ is called the '''pullback of $g$ along $f$''' and may be denoted $f^* g$.
\end{definition}