Image factorization in an abelian category ($A\twoheadrightarrow \mathrm{im}f\hookrightarrow B$)

For a morphism in an abelian category, its coimage $\mathrm{coker}(\ker f)$ is canonically isomorphic to its image $\ker (\mathrm{coker}f)$. This yields an epimorphism followed by a monomorphism, unique up to unique compatible isomorphism. Every epi-mono factorization has this same middle object.

Pull back the two arrows of an image factorization in an abelian category. Monomorphisms are preserved by pullback, and pullback stability of epimorphisms in an abelian category preserves its epic part. Pasting the squares gives the pulled-back composite. The resulting epi-mono factorization is therefore its image factorization by uniqueness.

A commuting square $vf=f^{\prime }u$ induces a unique map between their images by $I(u,v)p=p^{\prime }u$ and $i^{\prime }I(u,v)=vi$. The cokernel in a category property supplies existence, and cancellation of the epimorphic $p$ supplies uniqueness. These equations prove identity and composition laws, giving a functor from the arrow category to the abelian category.