Let $F:\mathcal{C}\to\mathcal{D}$ be a functor of small categories, and let $\mathcal{A}$ be a (co)complete category. The pre-composition with $F$ induces a functor $$F^\ast:\mathbf{Fun}(\mathcal{D},\mathcal{A})\to \mathbf{Fun}(\mathcal{C},\mathcal{A}).$$ A left (resp. right) Kan extension along $F$ is a left (resp. right) adjoint to $F^\ast$.
- Is there a sufficient and necessary conditions on $F$, so that the left and right Kan extensions coincide?
- What can one deduce when the left and right Kan extensions coincide?
Particularly, does it imply that $F^\ast$ is an equivalence of categories?