\begin{definition}[Definition:Coequalizer]
Let $\mathbf C$ be a metacategory.
Let $f, g: C \to D$ be morphisms with common domain and codomain.
An '''equalizer''' for $f$ and $g$ is a morphism $q: D \to Q$ such that:
:$q \circ f = q \circ g$
and subject to the following UMP:
:For any $z: D \to Z$ such that $z \circ f = z \circ f$, there is a unique $u: Q \to Z$ such that:
:::::$\begin{xy}\xymatrix{
 C
  \ar[r]<2pt>^*{f}
  \ar[r]<-2pt>_*{g}
& 
 D
  \ar[r]^*{q}
  \ar[rd]_*{z}
&
 Q
  \ar@{.>}[d]^*{u}
\\
& &
 Z
}\end{xy}$
:is a commutative diagram. I.e., $z = u \circ q$.
\end{definition}