The Stacks project

Proof. The statement about the étale topology is the content of Lemma 59.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the ph topology. By Topologies, Lemma 34.8.15 it suffices to show that given a proper surjective morphism $V \to U$ of schemes over $S$ we have an equalizer diagram

Set $\mathcal{G} = \pi _ S^{-1}\mathcal{F}|_{U_{\acute{e}tale}}$. Consider the commutative diagram

We have

where we use $f_*$ and $f^{-1}$ to denote functorialities between small étale sites. Second, we have

The two maps in the equalizer diagram come from the two maps

Thus it suffices to prove $\mathcal{G}$ is the equalizer of these two maps of sheaves. Let $\overline{u}$ be a geometric point of $U$. Set $\Omega = \mathcal{G}_{\overline{u}}$. Taking stalks at $\overline{u}$ by Lemma 59.91.4 we obtain the two maps

where $\underline{\Omega }$ indicates the constant sheaf with value $\Omega $. Of course these maps are the pullback by the projection maps. Then it is clear that the sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection, and sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection. The sections in the intersection of the images of these pullback maps are constant on all of $V_{\overline{u}} \times _{\overline{u}} V_{\overline{u}}$, i.e., these come from elements of $\Omega $ as desired. $\square$