The Proposition appearing in the wonderful answer of Zhen Lin to this other question states that for a small category $\mathbb{C}$ and an object $X\in \mathbb{C}$
(*)\begin{equation} \widehat{\mathbb{C}}\downarrow Y(X) \cong \widehat{\mathbb{C}\downarrow X} \end{equation}
i.e. the category of presheaves on $\mathbb{C}$ over the Yoneda embedded $X$ is equivalent (even isomorphic?) to the category of presheaves on $\mathbb{C}\downarrow X$. This appears also in Sheaves in Geometry and Logic by MacLane and Moerdijk, Exercise III.8.
There is an obvious functor $(-)|_X:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}\downarrow X}$ by precomposing a presheaf $\mathbb{C}^{op}\to Set$ with the opposite of the forgetful functor $\mathbb{C}\downarrow X\to \mathbb{C}$.
- Can this functor $(-)|_X$ be identified with the isomorphism $F$ from the left to the right in (*)? If not, how do $F$ and $(-)|_X$ relate?
If you start with an object $p:Z\to Y(X)$ of $\widehat{\mathbb{C}}\downarrow Y(X)$ and apply $(-)|_X$, you get $Z|_X\to Y(X)_X$ but $Y(X)_X=Hom_{\mathbb{C}\downarrow X}(-,X)$ is the terminal presheaf on $\mathbb{C}\downarrow X$. Hence it seems to me that the structure morphism $p$ is not respected by $(-)|_X$, we only get $Z|_X$.
On the other hand, the last page of chapter VII of the above mentioned book (or Exercise III.8.b) states the equivalence (*) for sheaves where $\mathbb{C}\downarrow X$ carries the obvious Grothendieck topology induced from $\mathbb{C}$. Doesn't this somehow suggest that the functor $(-)_X$ plays a role in the equivalence?