Consider a functor $F:\mathcal{A}\to \mathcal{B}$ between two small categories. Let $\mathcal{K}$ be a locally presentable category. Consider a functor $G:\mathcal{A}\to \mathcal{K}$, there is a natural transformation $G\Rightarrow (\mathrm{Lan}_F G)\circ F$ coming from the adjunction.

Is there a known necessary and sufficient condition for the natural transformation $G\Rightarrow (\mathrm{Lan}_F G)\circ F$ to be an isomorphism ?

By Proposition 3.7.3 of Borceux's book (Handbook of categorical algebra vol. 1), a sufficient condition is that $F:\mathcal{A}\to \mathcal{B}$ is full and faithful.

EDIT: by Exercice 3.9.5 of Borceux's book (Handbook of categorical algebra vol. 1), this condition is not necessary. Another sufficient condition is $F$ full and $\forall A,A'\in \mathcal{A}$ and any parallel arrows $f,f':A\rightrightarrows A', Gf=Gf' \Rightarrow Ff=Ff$ (there is a typo by the way in the book, $A$ is written instead of $A'$)