In a category with all kernel pairs and coequalisers of kernel pairs, the following conditions are equivalent:
- regular epimorphisms are stable under composition;
- regular epimorphisms coincide with strong epimorphisms;
- for any morphism $f$, if $m_f \circ e_f$ is its factorisation through the coequaliser of its kernel pair, $m_f$ is a monomorphism;
- regular epimorphisms and monomorphisms form a factorisation system.
This is proved in Monomorphisms, Epimorphisms, and Pull-backs by Kelly (Propositions 2.7 and 3.8). Note: Kelly takes as definition of regular epimorphism what is called elsewhere strict epimorphism, but these notions coincide when kernel pairs exist. (And Kelly doesn’t give the “factorisation system” version.)
Kelly gives an example of a pre-abelian category in which regular monomorphisms do not compose: the category of abelian groups with no elements of order 4 (last paragraph of p. 126). The dual category is a pre-abelian category in which regular epimorphisms do not compose.