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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.

Suppose a category has pullbacks and coequalizers and every pullback of a regular epimorphism is epic. Coequalizing the kernel pair of any morphism produces a factorization as a regular epimorphism followed by a monomorphism. Consequently every strong epimorphism is regular.

The coequalizer universal property supplies the diagonal filler against any monomorphism, so every regular epimorphism is strong. In a regular category, every strong epimorphism is regular: factor it as a regular epimorphism followed by a monomorphism and use the lifting property to make the monomorphism an isomorphism.

A functor between regular categories is regular if it preserves finite limits and regular epimorphisms. It then preserves image factorizations, since finite-limit preservation preserves monomorphisms. A representable functor $\mathcal{C}(P,-)$ is regular exactly when $P$ is projective with respect to regular epimorphisms. Regularity of a functor does not alone imply faithfulness or reflection of isomorphisms.

A regular monomorphism is a morphism that occurs as the equalizer of some parallel pair. Every regular monomorphism is a strong monomorphism.

This is the equalizer-defined subclass of monomorphisms.

A morphism $m:A\to B$ is strict when every $g:C\to B$ satisfying $hm=km\Rightarrow hg=kg$ for all parallel arrows $h,k$ out of $B$ factors uniquely through $m$. Every regular monomorphism is strict by its equalizer property. A strict monomorphism is strong: in a lifting square with an epimorphism, cancellation of the epimorphism makes the bottom arrow admissible, and its unique factorization gives the diagonal.

A strong epimorphism has the left lifting property with respect to every monomorphism. Thus every commutative square with the strong epimorphism on the left and a monomorphism on the right has a diagonal filler.

An object $S$ is subterminal when its unique morphism $S\to 1$ to the terminal object is a monomorphism. Equivalently, every object has at most one morphism to $S$. Subterminal objects form the preorder $\mathrm{Sub}(1)$.

In a regular category, a relation $R\hookrightarrow A\times B$ is total if its projection to $A$ is a regular epimorphism, and single-valued if two pairs with the same first component have the same second component. The latter makes that projection a monomorphism. A regular epimorphism that is monic is invertible, so $f={\pi }_B{\pi }_A^{-1}$ has graph exactly $R$. No axiom of choice or set-based element selection is used.

An object $A$ in a category with a terminal object is well-pointed when every monomorphism $m:B\to A$ through which all points $1\to A$ factor is invertible. This definition concerns detection of proper subobjects by points, and does not designate a chosen basepoint. If there are no points, it asserts that every subobject is the whole object. In particular, the empty set is well-pointed.