Image factorization ($A\twoheadrightarrow \mathrm{im}f\hookrightarrow B$)

An image of $f:A\to B$ is the least subobject $m:I\hookrightarrow B$ through which $f$ factors. In a category with pullback in a category constructions this gives a strong epimorphism–monomorphism factorization $f=me$: pull back any mono in a lifting square for $e$ to $I$; minimality of the image forces the resulting mono to be invertible, producing a diagonal. Conversely, the lifting property of the strong part proves minimality. No stability under pullback is asserted; that is an additional property in a regular category.

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.

In a category with pullback in a category constructions and image factorizations, direct image ${\exists }_f(A^{\prime })=\mathrm{im}(A^{\prime }\hookrightarrow A\stackrel{f}{\to }B)$ is a left adjoint to inverse image on subobjects. Frobenius reciprocity is ${\exists }_f(A^{\prime }\cap f^{\ast }{B}^{\prime })={\exists }_f(A^{\prime })\cap B^{\prime }$. It holds for all such subobjects exactly when strong epimorphisms are stable under pullback along monomorphisms. For sufficiency pull the strong part of the image factorization back along the mono into its intersection with $B^{\prime }$; for necessity take $A^{\prime }=A$ and $f$ strong, whose image is the whole codomain.