Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and suppose $F\colon\mathcal{C}\to\mathcal{D}$ is a functor. It induces two adjoint pairs between $Set~^{\mathcal{C}}$ and $Set~^{\mathcal{D}}$; one is denoted $(F^\star,F_\star)$ and one is denoted $(F_!,F^\star)$. One proves easily that the counit to $(F^\star,f_\star)$ is a natural isomorphism of functors $\mathcal{C}\to Set$ if and only if $F$ is fully faithful.

I am interested in the counit of the other adjunction $F_!:Set~^{\mathcal{C}}\Longleftrightarrow Set^{\mathcal{D}}:F^*$. Lets denote it by $$\epsilon_F\colon F_!F^*\to \operatorname {id}_{Set^{\mathcal{D}}}.$$

Question: Under what conditions on $F$ is $\epsilon_F^~$ a natural isomorphism?