Editing Coequalizer (section)

== Definition ==
A '''coequalizer''' is the [[colimit]] of a diagram consisting of two objects ''X'' and ''Y'' and two parallel [[morphism]]s {{nowrap|''f'', ''g'' : ''X'' → ''Y''}}.

More explicitly, a coequalizer of the parallel morphisms ''f'' and ''g'' can be defined as an object ''Q'' together with a morphism {{nowrap|''q'' : ''Y'' → ''Q''}} such that {{nowrap|1=''q'' ∘ ''f'' = ''q'' ∘ ''g''}}. Moreover, the pair {{nowrap|(''Q'', ''q'')}} must be [[universal property|universal]] in the sense that given any other such pair (''Q''&prime;, ''q''&prime;) there exists a unique morphism {{nowrap|''u'' : ''Q'' → ''Q''&prime;}} such that {{nowrap|1=''u'' ∘ ''q'' = ''q''&prime;}}. This information can be captured by the following [[commutative diagram]]:
<div style="text-align: center;">[[Image:Coequalizer-01.svg|x100px|class=skin-invert]]</div>


As with all [[universal construction]]s, a coequalizer, if it exists, is unique [[up to]] a unique [[isomorphism]] (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).

It can be shown that a coequalizing arrow ''q'' is an [[epimorphism]] in any category.