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.

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.