  On a category $\mathscr{C}$ with a designated (well-behaved) class
$\mathcal{M}$ of monomorphisms, a closure operator in the sense of D. Dikranjan
and E. Giuli is a pointed endofunctor of $\mathcal{M}$, seen as a full
subcategory of the arrow-category $\mathscr{C}^\mathbf{2}$ whose objects are
morphisms from the class $\mathcal{M}$, which "commutes" with the codomain
functor $\mathsf{cod}\colon \mathcal{M}\to \mathscr{C}$. In other words, a
closure operator consists of a functor $C\colon \mathcal{M}\to\mathcal{M}$ and
a natural transformation $c\colon 1_\mathcal{M}\to C$ such that $\mathsf{cod}
\cdot C=C$ and $\mathsf{cod}\cdot c=1_\mathsf{cod}$. In this paper we adapt
this notion to the domain functor $\mathsf{dom}\colon
\mathcal{E}\to\mathscr{C}$, where $\mathcal{E}$ is a class of epimorphisms in
$\mathscr{C}$, and show that such closure operators can be used to classify
$\mathcal{E}$-epireflective subcategories of $\mathscr{C}$, provided
$\mathcal{E}$ is closed under composition and contains isomorphisms.
Specializing to the case when $\mathcal{E}$ is the class of regular
epimorphisms in a regular category, we obtain known characterizations of
regular-epireflective subcategories of general and various special types of
regular categories, appearing in the works of the second author and his
coauthors. These results show the interest in investigating further the notion
of a closure operator relative to a general functor. They also point out new
links between epireflective subcategories arising in algebra, the theory of
fibrations, and the theory of categorical closure operators.
