The Stacks project

Lemma 61.18.2. Let $S$ be a scheme and let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point. The category of pro-étale neighbourhoods of $\overline{s}$ is cofiltered.

Proof. The proof is identitical to the proof of Étale Cohomology, Lemma 59.29.4 but using the corresponding facts about weakly étale morphisms proven in More on Morphisms, Lemmas 37.62.5, 37.62.6, and 37.62.13. $\square$

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