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 .
In a category with all finite limits and colimits, the image is defined as the equalizer of the so-called cokernel pair, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms , on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.
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 morphisms. (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 .