Lemma 61.12.9. Let S be a scheme. Let \mathit{Sch}_{pro\text{-}\acute{e}tale} be a big pro-étale site containing S. Both S_{pro\text{-}\acute{e}tale} and (\textit{Aff}/S)_{pro\text{-}\acute{e}tale} are sites.
Proof. Let us show that S_{pro\text{-}\acute{e}tale} is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale} is a site, it suffices to prove that given any covering \{ U_ i \to U\} of (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale} with U \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale}) we also have U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale}). This follows from the definitions as the composition of weakly étale morphisms is weakly étale.
To show that (\textit{Aff}/S)_{pro\text{-}\acute{e}tale} is a site, reasoning as above, it suffices to show that the collection of standard pro-étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This follows from Lemma 61.12.2 and the corresponding result for standard fpqc coverings (Topologies, Lemma 34.9.11). \square
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