Finite-limit-preserving functor

A finite-limit-preserving functor preserves a terminal object and pullbacks, equivalently all finite limits. It is also called left exact. Such functors from a Cartesian syntactic category to a finite-limit category correspond to models of the associated Cartesian theory.

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.