Let $C$ be a category and let $F:C\rightarrow D$ be a functor with $D$ locally presentable and cartesian closed. When does the Yoneda extension $\widehat{F}=Lan\_{y} F:[C^{op},Set]\rightarrow D$ preserve finite products?



Yoneda extension preserving finite products?
(Update: the following answer applies only in the case that $C$ has finite products.)


The left Kan extension $\hat{F}$ preserves finite products just when $F$ does. 


One direction is easy since $F \cong \hat{F} \circ Y$ and $Y$ preserves finite products.


The converse is an old result of Borceux and Day from their paper [On finite product preserving Kan extensions.](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700010455) in the Bulletin of the Australian Mathematical Society. Combining Theorem 1.5 with Example 3.1 of that paper 
gives the result. 


(Warning: they do everything in maximal generality and in the enriched setting so their results may take some time to parse.)


