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] ^*{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., $z = u \circ q$.

Also see

 * Equalizer, the dual notion