The Stacks project

Lemma 73.4.8. Let $S$ be a scheme. Let $X$ be an object of $(\textit{Spaces}/S)_{\acute{e}tale}$. The inclusion functor $X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites

\[ \pi _ X : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow X_{spaces, {\acute{e}tale}} \]

and a morphism of topoi

\[ i_ X : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \]

such that $\pi _ X \circ i_ X = \text{id}$. Moreover, $i_ X = i_{\text{id}_ X}$ with $i_{\text{id}_ X}$ as in Lemma 73.4.7. In particular the functor $i_ X^{-1} = \pi _{X, *}$ is described by the rule $i_ X^{-1}(\mathcal{G})(U/X) = \mathcal{G}(U/X)$.

Proof. In this case the functor $u : X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 73.4.7 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8. $\square$

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