= Flat functor and finite-limit preservation
{title2=$\mathcal C\text{ finitely complete}\Longrightarrow(F\text{ flat}\iff F\text{ left exact})$}
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.
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