The concept of coequalizer in a general category is the generalization of the construction where for two functions $f,g$ between sets $X$ and $Y$
one forms the set $Y/_\sim$ of equivalence classes induced by the equivalence relation generated by the relation
for all $x \in X$. This means that the quotient function $p \colon Y \longrightarrow Y/_\sim$ satisfies
(a map $p$ satisfying this equation is said to “co-equalize” $f$ and $g$) and moreover $p$ is universal with this property.
In this form this may be phrased generally in any category.
In some category $\mathcal{C}$, the coequalizer $coeq(f,g)$ of two parallel morphisms $f$ and $g$ between two objects $X$ and $Y$ is (if it exists), the colimit under the diagram formed by these two morphisms
Equivalently:
In a category $\mathcal{C}$ a diagram
is called a coequalizer diagram if
$p \circ f = p \circ g$;
$p$ is universal for this property: i.e. if $q \colon Y \to W$ is a morphism of $\mathcal{C}$ such that $q \circ f = q \circ g$, then there is a unique morphism $q' \colon Z \to W$ such that $q' \circ p = q$
By formal duality, a coequalizer in $\mathcal{C}$ is equivalently an equalizer in the opposite category $\mathcal{C}^{op}$.
Morphisms that coequalize some pair of parallel morphisms are called regular epimorphisms.
In any category:
If a morphism
is the coequalizer of some pair of parallel morphisms (hence: is a regular epimorphism)
has a kernel pair,
then it is also the coequalizer of its kernel pair.
If a kernel pair has a coequalizer, then it is the kernel pair of its coequalizer.
Coequalizers are closely related to pushouts:
A diagram
is a coequalizer diagram, def. , precisely if
is a pushout diagram.
Conversely:
A diagram
is a pushout square, precisely if
is a coequalizer diagram.
For $\mathcal{C} =$ Set, the coequalizer of two functions $f$, $g$ is the quotient set by the equivalence relation generated by the relation $f(x) \sim g(x)$ for all $x \in X$.
For $\mathcal{C} =$ Top, the coequalizer of two continuous functions $f$, $g$ is the topological space whose underlying set is the quotient set from example , and whose topology is the corresponding quotient topology.
Coequalizers were defined in the paper
for any finite collection of parallel morphisms. The paper refers to them as right equalizers, whereas equalizers are referred to as left equalizers.
Textbook accounts:
Francis Borceux, Section 2.4 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)
Paul Taylor, Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999 (webpage)