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

and a morphism of topoi

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 $i_ X^{-1}(\mathcal{G})(U/X) = \mathcal{G}(U/X)$.

## Comments (0)

There are also: