In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category.
A coequalizer is the colimit of a diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y.
As with all universal constructions, 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.
In preadditive categories it makes sense to add and subtract morphisms (the hom-sets actually form abelian groups).