The Stacks project

Proof. Choose an equivalence $j : \mathcal{S}_ F \to \mathcal{X}$, see Algebraic Stacks, Sections 94.7 and 94.8. An object of $F_{\acute{e}tale}$ is a scheme $U$ together with an étale morphism $\varphi : U \to F$. Then $\varphi $ is an object of $\mathcal{S}_ F$ over $U$. Hence $j(\varphi )$ is an object of $\mathcal{X}$ over $U$. In this way $j$ induces a functor $u : F_{\acute{e}tale}\to \mathcal{X}$. It is clear that $u$ is continuous and cocontinuous for the étale topology on $\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully faithful. Also, fibre products and equalizers exist in $F_{\acute{e}tale}$ and $u$ commutes with them because these are computed on the level of underlying schemes in $F_{\acute{e}tale}$. Thus Sites, Lemmas 7.21.5, 7.21.6, and 7.21.7 apply. In particular $u$ defines a morphism of topoi $i_ F : \mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ and there exists a left adjoint $i_{F, !}$ of $i_ F^{-1}$ which commutes with fibre products and equalizers.

We claim that $i_{F, !}$ is exact. If this is true, then we can define $\pi _ F$ by the rules $\pi _ F^{-1} = i_{F, !}$ and $\pi _{F, *} = i_ F^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_{F, !}$ is right exact and preserves fibre products. Hence it suffices to show that $i_{F, !}* = *$ where $*$ indicates the final object in the category of sheaves of sets. Let $U$ be a scheme and let $\varphi : U \to F$ be surjective and étale. Set $R = U \times _ F U$. Then

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (F_{\acute{e}tale})$. Using the right exactness of $i_{F, !}$, using $i_{F, !} = (u_ p\ )^\# $, and using Sites, Lemma 7.5.6 we see that

is a coequalizer diagram in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$. Using that $j$ is an equivalence and that $F = U/R$ it follows that the coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ of the two maps $h_{u(R)} \to h_{u(U)}$ is $*$. We omit the proof that these morphisms are compatible with structure sheaves. $\square$