It ought to be true when E is a (Grothendieck) topos (prove it for presheaf categories and then for left-exact-reflective subcategories). Conversely, if E is cocomplete and finitely complete, has the stated property (i.e.the Yoneda extension of any left exact functor from a finitely complete category into E is left exact), and moreover has a small dense subcategory C, then if C’ denotes the closure of C in E under finite limits, the Yoneda extension of the inclusion C’→E must be a left exact reflection of the embedding of E into Psh(C’), so that E is a topos.

In fact, even if instead of having a small dense subcategory we assume that E is *totally cocomplete*, meaning that its own Yoneda embedding E→Psh(E) has a left adjoint, then the stated property implies that that left adjoint is left exact, i.e. E is “lex total.” And as long as ob(E) is no bigger than ob(Set), then lex-totality in fact also implies that E is a topos. So morally, at least, I think the extra condition is just “being a topos.”

At flat functor there is a statement that a functor $F : C \to \mathcal{E}$ on a complete category to a cocomplete category is left exact precisely if its Yoneda extension is.

I know this for $\mathcal{E} = Set$. There must be some extra conditions on $\mathcal{E}$ that have to be mentioned here.

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