Pullback of sieve along functor | Higher Topos Theory notes

Notation 6.24.

Given an object $X \in C$ and a sieve $U \hookrightarrow y(u(X))$ of $$u(X)$$ in $$D$$ , we define its pullback along $$u$$ as the sieve $u^{-1}(U) \hookrightarrow y(X)$ of $$X$$ obtained by forming the following pullback square in $\PSh(C)$ :

here the bottom map is the unit map $y(X) \to u^*u_!y(X) = u^*(y(u(X)))$ . Note that $u^{-1}(U)$ consists of those morphisms $Z \to X$ in $$C$$ for which the map $u(Z) \to u(X)$ lies in the sieve $$U$$ .