TITLE: Is composition of regular epimorphisms always regular?
QUESTION [8 upvotes]: In a finitely complete and cocomplete category. Does it always hold that the composition of two regular epimorphisms is regular? And if it's not the case, what kind of additional constraints can make it true (say, a pre-abelian category)?
What I already knew is it holds for categories where regular epimorphisms and strong epimorphisms conincide.

REPLY [6 votes]: My standard example of a category where regular epimorphisms are not closed under composition is the category $\mathbf{Cat}$ of small categories.
Let $\mathbb{2}=\{0\to 1\}$ be the category with two objects and one non-identity morphism between them, and let $F:\mathbb{2}\to\mathbb{N}$ be the functor sending this morphism to $1$, where $\mathbb{N}$ is the additive monoid of natural numbers, viewed as 1-object category.
Let $G:\mathbb{N}\to\mathbb{Z}$ be the inclusion of additive monoids, viewed as functor between the associated 1-object categories, and let $H: \mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}$ be the quotient map, again viewed as functor between 1-object categories.
Then $F$ and $H\circ G$ are regular epis in $\mathbf{Cat}$, but $H\circ G\circ F$ is not.