Left Kan extension of a (co)presheaf along a fully-faithful functor

Question

Let $p: C \to D$ be a fully-faithful functor, let $F: D \to \mathbf{Set}$ be any functor. Is it true that the left Kan extension $\operatorname{Lan}_p Fp$ is given by $$ (\operatorname{Lan}_p Fp)(X) = \{ Ff(y) \in F(X) \mid y \in Fp(Y) \text{ and } f: p(Y) \to X \}. $$ Treating $C$ as a subcategory of $D,$ left Kan extension is the subfunctor of $F$ generated by elements of $F$ in $C$.

This is equivalent to showing the canonical natural transformation $\operatorname{Lan}_p Fp \to F$ is monic, but I have no idea how to do this.

I believe this is true, as this holds for some simple examples. A nontrivial example is the skeleton of a simplicial set.

Answer 1

It is not true. The point is that elements of $F$ generated by elements of $F p$ may have additional relations not generated by $F p$. The simplest example is the case where $D = \{ c \to d \leftarrow c' \}$, $C = \{ c, c' \}$, $p$ is the inclusion, and $F$ is the constant functor with value $1$. Then $\operatorname{Lan}_p (F p) (d)$ has two elements, so $\operatorname{Lan}_p (F p) \Rightarrow F$ is not monic.