I'm guessing the answer to this question is well-known:


Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}\_Y\left(F\right).$ Under what conditions does $\mathbf{Lan}\_Y\left(F\right)$ preserve colimits? Notice that if $C=P$ and $Y=id\_C,$ then $\mathbf{Lan}\_Y\left(F\right)=F,$ so this is not true in general. Would $F$ preserving colimits imply this?


Dually, under what conditions does a right Kan extension preserve limits?


Thank you.



When do Kan extensions preserve limits/colimits?
The pointwise left Kan extension of F along Y is a coend of functors $Lan\_{Y}(F) = \int^{x}P(Yx,-).Fx$ where each functor $P(Yx,-).Fx$ is the composite of the representable $P(Yx,-):P \to Set$ and the copower functor $(-.Fx):Set \to D$. As a coend (colimit) of the $P(Yx,-).Fx$ the left Kan extension preserves any colimit by each of these functors.


Now the copower functor $(-.Fx)$ is left adjoint to the representable $D(Fx,-)$ and so preserves all colimits, so that $P(Yx,-).Fx$ preserves any colimit preserved by $P(Yx,-)$.
Therefore $Lan\_{Y}(F)$ preserves any colimit preserved by each representable $P(Yx,-):P \to Set$ for $x \in C$.


If Y is the Yoneda embedding we have $P(Yx,-)=[C^{op},Set](Yx,-)=ev\_{x}$ the evaluation functor at x which preserves all colimits, so that left Kan extensions along Yoneda preserve all colimits. 


Or if each $P(Yx,-)$ preserves filtered colimits then left Kan extensions along Y preserve filtered colimits.


I think this is all well known but don't know a reference.


