Flat functor and finite-limit preservation ($\mathcal{C}\text{ finitely complete}⟹(F\text{ flat}⟺F\text{ left exact})$)

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.