nLab
coequalizer

Definition

In a category $C$ a diagram of morphisms of $C$

$$U\underset{b}{\overset{a}{\rightrightarrows }}V\stackrel{c}{\to }X$$U \underoverset{b}{a}{\rightrightarrows} V \overset{c}{\rightarrow} X

is called a coequalizer diagram if

1. $ca=cb$; and
2. $c$ is universal for this property: i.e. if $f:V\to Y$ is a morphism of $C$ such that $fa=fb$, then there is a unique morphism $f\prime :X\to Y$ such that $f\prime c=f$.

This concept is a special case of that of colimit; specifically, it’s the colimit of the diagram

$$U\underset{b}{\overset{a}{\rightrightarrows }}V.$$U \underoverset{b}{a}{\rightrightarrows} V .

A coequalizer in $C$ is an equalizer in the opposite category $C^{\mathrm{op}}$.