For a general (orthogonal) factorization system , the factorizations show that for all objects the full inclusion (where consists of morphisms in with target ) has a left adjoint, hence is a reflective subcategory.
The analogous statement also holds in (∞,1)-category theory, or rather at least in locally cartesian closed (∞,1)-categories. A discussion of this and formal proof in terms of homotopy type theory is in (Shulman).
A stable reflective factorization system is sometimes called local.
More generally, in an (∞,1)-topos for all the (n-epi, n-mono) factorization system (see there for more details) is a stable orthogonal factorization system in an (∞,1)-category.
The relation between stable factorization systems and the Beck-Chevalley condition of the associated fibrations is discussed in
The notion appears also for instance in
Max Kelly, A note on relations relative to a factorization system, Lecture Notes in Mathematics, 1991, Volume 1488 (1991)
Stefan Milius, Relations in categories, PhD thesis (pdf)
Discussion of epimorphisms in toposes is for instance in