Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some reasons we know that both the left and the right Kan extensions of $F$ along $Y$ exist.
Question: When are they cocontinuous?
I know that if $Y$ is the Yoneda embedding, then the left Kan extension is cocontinuous, but this is not a case I am interested in, since the Yoneda embedding is not cocontinuous in general.
I also know from this question that $Lan_Y(F)$ preserves any colimit preserved by each representable functor $B(Yx,−): B \to \mathsf{Set}$ for $x \in A$, but it seems to me that in the context I am interested in, this is not the case.
I am not familiar with these constructions, can anyone give me some hints, please?
In the particular situation I am interested in $Y:A\to \left(\mathsf{Set}^A\right)^{op}$ is the "covariant Yoneda embedding".
