Regular functor
= Regular functor
A <functor> between <regular categories> is regular if it preserves <finite limits> and <regular epimorphisms>. It then preserves <image factorizations>, since finite-limit preservation preserves <monomorphisms>. A <representable functor> $\mathcal C(P,-)$ is regular exactly when $P$ is projective with respect to regular epimorphisms. Regularity of a functor does not alone imply faithfulness or reflection of isomorphisms.