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.
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