It is well known that for ordinary categories, if $C$ has finite limits and $D$ is cocomplete, and 
$A:C \to D $ is left-exact (i.e. preserves finite limits) then the left-Kan extension of $F$ along the Yoneda embedding $y:C \hookrightarrow Set^{C^{op}}$ is left-exact. I'm pretty sure this is still true for $\left(\infty,1\right)$-categories, once we replace the role of presheaves with that of $\infty$-presheaves, but is this written up somewhere?



Flatness for infinity functors
For reference, at least when $D$ is an infinity topos, which I believe is probably necessary, this is Proposition 6.1.5.2 in HTT.


