Let me first recap the definitions of flatness, using the nLab's terminology:
A functor into $F : \mathcal{C} \to \mathbf{Sets}$ is Set-valued flat when the category of elements $\int E$ is filtered. This implies, and is implied by, left exactness of the Kan extension $\mathrm{Lan}_{よ}(F) : [\mathcal{C}^{\mathrm{op}}, \mathbf{Sets}]$. A functor $F : \mathcal{C} \to \mathcal{E}$ is representably flat when each $\mathrm{Hom}_E(B, F(-)) : \mathcal{C} \to \mathbf{Sets}$, for $B$ ranging over the objects of $\mathcal{E}$, is Set-valued flat.
The nLab claims that for a representably flat functor $F : \mathcal{C} \to \mathcal{E}$, where both categories are small, the extension of $F$ to a functor $[\mathcal{C}^{\mathrm{op}}, \mathbf{Sets}] \to [\mathcal{E}^{\mathrm{op}}, \mathbf{Sets}]$ preserves finite limits. I'm interested in a different setup: If $\mathcal{E}$ is finitely complete and cocomplete, does representable flatness of $F : \mathcal{C} \to \mathcal{E}$ ensure that $\mathrm{Lan}_{よ}(F) : [\mathcal{C}^\mathrm{op}, \mathbf{Sets}] \to \mathcal{E}$ is left exact? If this is not the case for an arbitrary cocomplete category $\mathcal{E}$, does the answer change when $\mathcal{E}$ is a sheaf topos?
Best Answer
I assume you mean that $\mathcal E$ is not only cocomplete, but also finitely complete (otherwise I'm not sure what you would mean by a left-exact functor with codomain $\mathcal E$). Then Garner and Lack's paper Lex Colimits seems to answer part of your question ; proposition 2.4(5) and 2.5 combined assert that :
Given a cocomplete and finitely complete category $\mathcal E$, the following are equivalent :
Given that any lex functor $\mathcal D \to \mathcal E$ is representably flat, it follows that your $\mathcal E$ needs to be an infinitary pretopos at least.