Editing Coequalizer (section)

== Special cases ==
In categories with [[zero morphism]]s, one can define a ''[[cokernel]]'' of a morphism ''f'' as the coequalizer of ''f'' and the parallel zero morphism.

In [[preadditive category|preadditive categories]] it makes sense to add and subtract morphisms (the [[hom-set]]s actually form [[abelian group]]s). In such categories, one can define the coequalizer of two morphisms ''f'' and ''g'' as the cokernel of their difference:
: coeq(''f'', ''g'') = coker(''g'' – ''f'').

A stronger notion is that of an '''absolute coequalizer''', this is a coequalizer that is preserved under all functors.
Formally, an absolute coequalizer of a pair of parallel arrows {{nowrap|''f'', ''g'' : ''X'' → ''Y''}} in a category ''C'' is a coequalizer as defined above, but with the added property that given any functor {{nowrap|''F'' : ''C'' → ''D''}}, ''F''(''Q'') together with ''F''(''q'') is the coequalizer of ''F''(''f'') and ''F''(''g'') in the category ''D''. [[Split coequalizer]]s are examples of absolute coequalizers.