I am reading these lecture notes, and I am stuck on a crucial step. Let $F$ be a presheaf on a site $(\mathsf{C},J)$. In Lemma 24, I believe Lurie is claiming the following: if $R \subseteq S$ are two covering sieves on an object $X \in \mathsf{C}$, then $$\mathsf{Pre(C)}(R,F) = \lim_{U \xrightarrow{g} X \in S} \mathsf{Pre(C)}(g^*R, F),$$ where I am using the fact that $\mathsf{Pre(C)}(R,F) = \lim_{U \xrightarrow{g} X \in R} F(U)$ in the translation between Lurie's notation and mine. Recall that $g^*R = rU \times_{rX} R$, where $rU$ is image of the yoneda embedding on $U$. I don't see why this equality is true.
This would be a consequence of $R \cong \underset{U \xrightarrow{g} X \in S}{\text{colim}} g^* R$, but I don't see how to prove it, or if it is even true. Any guidance here would be very helpful.