Flat functor ($F:\mathcal{C}\to \mathcal{E}$)

A covariant functor into a Grothendieck topos is flat when its tensor extension from presheaves preserves finite limits. For set values, its category of elements with arrows carrying source elements to target elements is cofiltered; equivalently the functor is a filtered colimit of covariant representables. If the indexing category has finite limits, flatness is equivalent to preservation of finite limits. For a site, cover-to-joint-epimorphism continuity adds the condition needed by the Diaconescu equivalence for geometric morphisms.

For a small category with finite limits, a set-valued functor preserving them has a category of elements with finite limits, so its opposite is a filtered category. The covariant density presentation is therefore a filtered colimit of representable functors. Conversely, representable functors preserve categorical limits, and filtered colimits commute with finite limits in sets, proving the criterion.