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}

<8em,0em>*+{Q} = "E", <0em,0em>*+{C}  = "C", <4em,0em>*+{D}  = "D", <8em,-4em>*+{Z} = "A",

"C"+/^.3em/+/r.5em/;"D"+/^.3em/+/l.5em/ **@{-} ?>*@{>} ?*!/_.8em/{f}, "C"+/_.3em/+/r.5em/;"D"+/_.3em/+/l.5em/ **@{-} ?>*@{>} ?*!/^.8em/{g}, "D";"E" **@{-} ?>*@{>} ?*!/_.8em/{q}, "E";"A" **@{.} ?>*@{>} ?*!/_.8em/{u}, "D";"A" **@{-} ?>*@{>} ?*!/^.8em/{z}, \end{xy}$


 * is a commutative diagram. I.e., $z = u \circ q$.

Also see

 * Equalizer, the dual notion