Let $C$ be a category, $D$ be a Grothendieck topos, and suppose we have a fully faithful, left-exact functor $F:C\rightarrow D$. Let $Lan\_{y}F:PShv(C)\rightarrow D$ be the Yoneda extension of $F$. Since both $C,D$ are locally presentable and $Lan\_{y}F$ is cocontinuous, $Lan\_{y}F$ has a right adjoint, call it $G$. Is there an abstract way of seeing if $G$ is fully faithful?



Faithfulness of Right adjoint to Kan extension
Assuming $C$ is a small category, the right adjoint to $Lan\_y(F): PShv(C) = Set^{C^{op}} \to D$ is just the functor $D \to Set^{C^{op}}: d \mapsto \hom\_D(F-, d)$; this basic "[generalized nerve](https://ncatlab.org/nlab/show/nerve+and+realization)" construction traces back to the original 1958 paper by Kan where adjoint functors were first introduced: 


* Daniel Kan, *Functors involving c.s.s complexes*, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330--346 ([jstor](http://www.jstor.org/stable/1993103)).


Meanwhile, the condition that the nerve $N\_F$ defined by $N\_F(d) = \hom\_D(F-, d)$ is fully faithful (as in the case of the classical nerve induced by the inclusion $\Delta \hookrightarrow Cat$) is much considered in the categorical literature: we say that $F$ is *dense*, since it amounts to every object of $D$ being a colimit of objects coming from $C$ in a canonical way. A pretty good set of references is given in the [nLab article](https://ncatlab.org/nlab/show/dense+functor) on dense functors; some of the older references (such as by Isbell under the name "adequacy") pay particular attention to the case where $F$ itself is an embedding of a full subcategory. 


