Image and coimage in an abelian category ($\mathrm{im}f,\mathrm{coim}f$)

For $f:A\to B$ in an abelian category,The canonical morphism $\mathrm{coim}f\to \mathrm{im}f$ is an isomorphism, giving the canonical epimorphism--isomorphism--monomorphism factorization of $f$.

In an abelian category, the coimage of a morphism is its domain modulo its kernel in a category, expressed as a cokernel in a category. The canonical arrow from the coimage to the image is an isomorphism, giving the image factorization in an abelian category.