 Left-exact reflective subcategory of a regular category

Every left-exact reflective subcategory of a regular category is regular. The reflector sends a regular-epimorphism--monomorphism factorization to such a factorization among fixed objects. A morphism between fixed objects is a regular epimorphism precisely when the closure of its image under the closure operation induced by a left-exact reflector is the whole codomain; pullback stability follows from pullback stability of images and of closure.