= Flat functor
{title2=$F:\mathcal C\to\mathcal E$}
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>.
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