Intuitively, an absolute coequalizer is a diagram that is a coequalizer “purely for diagrammatic reasons.” The most common example is a split coequalizer. A trivial example of an absolute coequalizer that is not split is a diagram of the form
whenever is not a split epimorphism.
In fact, split coequalizers and “trivial” absolute coequalizers are the cases and of a general characterization of absolute coequalizers, which we now describe. Suppose that
is an absolute coequalizer. Then it must be preserved, in particular, by the hom-functor ; that is, we have a coequalizer diagram
in . In particular, that means that is surjective, and so in particular there exists some such that . In other words, is split epic.
Now the given coequalizer must also be preserved by the hom-functor , so we have another coequalizer diagram
in . We also have two elements and in with the property that (since ).
However, a coequalizer of two functions in is constructed as the quotient set of by the equivalence relation generated by the image of . That means that we set iff there is a finite sequence of elements of and a finite sequence with , such that , , and . We consider as the case .
Therefore, since and are in the same class of the equivalence relation on generated by and , they must be related by such a finite chain of elements of . That is, we must have morphisms and a sequence of binary digits such that , , and . (Note that if then this says precisely that we have a split coequalizer, and if it is the trivial case above.) Conversely, it is easy to check that given and satisfying these equations, the given fork must be a coequalizer, for essentially the same reason that any split coequalizer is a coequalizer. Thus we have a complete characterization of absolute coequalizers.
This characterization is essentially a special case of the characterization of absolute colimits (in unenriched categories).