It is well known that for ordinary categories, if $C$ has finite limits and $D$ is cocomplete, and $A:C \to D $ is leftexact (i.e. preserves finite limits) then the leftKan extension of $F$ along the Yoneda embedding $y:C \hookrightarrow Set^{C^{op}}$ is leftexact. 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?
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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. 

