Let $F\colon C \to D$ be a functor. The Kan Extension of $y\_D \circ F$ along $y\_C$ yields a functor $F\_!: Fun(C^{op},Set) \to Fun(D^{op},Set)$. Here, $y\_C$ and $y\_D$ denotes the respective Yoneda embeddings. It is well-known that if $C$ and $D$ have finite limits and $F$ preserves them, then so does $F\_!$.


**Question**: Suppose that we only assume that $C$ has pullbacks and that $F$ preserves them (so some pullbacks exist in $D$, too). Is it true that $F\_!$ preserves pullbacks as well?


(I'm willing to assume that both $C$ and $D$ have finite products, but in my case, the functor $F$ does certainly not preserve these.)


I went through Lurie's ∞-categorical version of the "well-known" part (6.1.5.2 in HTT), but I was not able to verify that the last step works without assuming that $F$ preserves products because this step involves some heavy machinery. I could give a different proof for this last step if $F\_!$ preserved monomorphisms, but I believe that this is not the case in general.



Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks
If you are wiling to assume that $C$ has a terminal object $1 \in C$, which I assume is the case as you said all finite products, you can do the following:


(As it is not clear if you are interested in a $1$-categorical statement or an $\infty$-categorical statement I'll write the proof in a very formal style which should accomodate $\infty$-categories as well with a little bit of work)


Saying that $F:C \rightarrow D$ preserve fiber products means that $F\_{/1}: C \rightarrow D\_{/F(1)}$ preserves all finite limits (as it preserve pullback and the initial objects).


In particular the induced functor $Prsh(C) \rightarrow Prsh(D\_{/F(1)}) $ preserves all finite limits by your "well known fact".


But $Prsh(D\_{/F(1)})$ identifies canonically with $Prsh(D)\_{/y\_D(F(1))}$ and $y\_D(F(1))$ is $F\_!(y\_C(1))$ with $y\_C$ being the terminal object of $Prsh(C)$.


In the end the functor: $F\_!$ preserve all finite limits when seen as a functor from $Prsh(C)$ to $Prsh(D)\_{/F\_!(1)}$ but this means that $F\_!$ preserve fiber product as asked.




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I have ignored the fact that $D$ does not have all finite limits as, as far as I know, this has never been part of the assumption of your "well known fact".


Indeed, if $C$ has all finite limits and $F:C \rightarrow D$ preserve them, then in order to check that $F\_!$ preserve finite limits one justs need to check that for all $d \in D$ the functor $X \mapsto Hom(d,F\_! ( X))$ from $Prsh(C)$ to $Set$ preserves all finite limits.


This functor is the left Kan extention along the Yoneda embedings of $C$ of the functor $C \rightarrow Set$ which send $c$ to $Hom(d,F(c))$, which commutes to finite limits by definition.


So in the end the relevant fact is that if $C$ has finite limits $F:C \rightarrow Set$ preserves them then its left Kan extention $Prsh(C) \rightarrow Set$ also preserve finite limits. But existence of limits in $D$ never really played a role.




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I'm adding a counter example to show that the assumption that $C$ has a terminal object cannot be completely relaxed.


Let $C$ be a group $G$, seen as a one object category. And let $D$ be the terminal category. With $F:C \rightarrow D$ the unique functor.


The category $C$ have no terminal object but it has all fibered product (because all map are isomorphisms), and the functor $F$ preserve all fibered product limits (in fact we have both that any functor out of $C$ and that any functor to $D$ preserves all fibered product)


$F\_!$ is the functor from $G$-set to set which send any $G$-set to its quotient $X/G$.


$F\_!$ send the terminal object to the terminal object but does not preserve product: if $G$ denotes $G$ with its multiplication action on itself then $F\_! (G \times G) = G$ but $F\_!(G)=\{\*\}$.


So $F\_!$ does not preserve fibered product.


Two remarks: 


* I do not know if there are other condition than having a terminal object that allow this sort of result to work.
* This examples suggest that things might work better if one work with $\infty$-categories and $Prsh$ means space valued presheaves. I do not know the status of this (**Edit:** see me second answer about this specific point).


