# Definition:Coequalizer

## Definition

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 \[email protected]{.>}[d]^*{u} \\ & & Z }\end{xy}$
is a commutative diagram. I.e., $z = u \circ q$.