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Proposition: If has all equalizers then the in the factorization of (1) is an epimorphism.
Let be such that , one needs to show that . Since the equalizer of exists, factorizes as with monic. But then is a factorization of with monomorphism. Hence by the universal property of the image there exists a unique arrow such that and since is monic . Furthermore, one has and by the monomorphism property of one obtains .
This means that and thus that equalizes , whence .
Finite bicompleteness of the category ensures that pushouts and equalizers exist.
can be called regular image as is a regular monomorphism, i.e. the equalizer of a pair of morphism. (Recall also that an equalizer is automatically a monomorphism).
In an abelian category, the cokernel pair property can be written and the equalizer condition . Moreover, all monomorphisms are regular.
Theorem — If always factorizes through regular monomorphisms, then the two definitions coincide.
First definition implies the second: Assume that (1) holds with regular monomorphism.
Equalization: one needs to show that . As the cokernel pair of and by previous proposition, since has all equalizers, the arrow in the factorization is an epimorphism, hence .
Universality: in a category with all colimits (or at least all pushouts) itself admits a cokernel pair
Moreover, as a regular monomorphism, is the equalizer of a pair of morphisms but we claim here that it is also the equalizer of .
Indeed, by construction thus the "cokernel pair" diagram for yields a unique morphism such that . Now, a map which equalizes also satisfies , hence by the equalizer diagram for , there exists a unique map such that .
Finally, use the cokernel pair diagram (of ) with : there exists a unique such that . Therefore, any map which equalizes also equalizes and thus uniquely factorizes as . This exactly means that is the equalizer of .
Second definition implies the first:
Factorization: taking in the equalizer diagram ( corresponds to ), one obtains the factorization .
Universality: let be a factorization with regular monomorphism, i.e. the equalizer of some pair .
Then so that by the "cokernel pair" diagram (of ), with , there exists a unique such that .
Now, from (m from the equalizer of (i1, i2) diagram), one obtains , hence by the universality in the (equalizer of (d1, d2) diagram, with f replaced by m), there exists a unique such that .