The Stacks project

Proof. Recall that $\mathcal{F}^\#$ is equal to $(\mathcal{F}^+)^+$, see Sites, Theorem 7.10.10, where $\mathcal{F}^+$ is the presheaf which sends an object $U$ of $X_{pro\text{-}\acute{e}tale}$ to $\mathop{\mathrm{colim}}\nolimits H^0(\mathcal{U}, \mathcal{F})$ where the colimit is over all pro-étale coverings $\mathcal{U}$ of $U$. Thus it suffices to prove that (a) $\mathcal{F}^+$ sends finite disjoint unions to products and (b) sends $W$ to $\mathcal{F}(W)$. If $U = U_1 \amalg U_2$, then given a pro-étale covering $\mathcal{U} = \{ f_ j : V_ j \to U\}$ of $U$ we obtain pro-étale coverings $\mathcal{U}_ i = \{ f_ j^{-1}(U_ i) \to U_ i\}$ and we clearly have

because $\mathcal{F}$ sends finite disjoint unions to products (this includes the condition that $\mathcal{F}$ sends the empty scheme to the singleton). This proves (a). Finally, any pro-étale covering of $W$ can be refined by a finite disjoint union decomposition $W = W_1 \amalg \ldots W_ n$ by Lemma 61.13.2. Hence $\mathcal{F}^+(W) = \mathcal{F}(W)$ exactly because the value of $\mathcal{F}$ on $W$ is the product of the values of $\mathcal{F}$ on the $W_ j$. This proves (b). $\square$

Proof. Recall that $f_{small}^{-1}\mathcal{F}$ is the sheaf associated to the presheaf

on $X_{\acute{e}tale}$, see Sites, Sections 7.14 and 7.13; we've suppressed from the notation that the colimit is over the opposite of the category $\{ U \to V, V \in Y_{pro\text{-}\acute{e}tale}\}$. By Lemma 61.16.1 it suffices to prove that $u_ p\mathcal{F}$ sends finite disjoint unions to products. Suppose that $U = U_1 \amalg U_2$ is a disjoint union of open and closed subschemes. There is a functor

which is initial (Categories, Definition 4.17.3). Hence the corresponding functor on opposite categories is cofinal and by Categories, Lemma 4.17.2 we see that $u_ p\mathcal{F}$ on $U$ is the colimit of the values $\mathcal{F}(V_1 \amalg V_2)$ over the product category. Since $\mathcal{F}$ is a sheaf it sends disjoint unions to products and we conclude $u_ p\mathcal{F}$ does too. $\square$

Proof. Observe that we have a morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ such that $\pi _ S \circ i_ f = f_{small}$ as morphisms $T_{pro\text{-}\acute{e}tale}\to S_{pro\text{-}\acute{e}tale}$, see Lemma 61.12.12. Since pullback is transitive we see that $i_ f^{-1} \pi _ S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$ as desired.

Let $\{ g_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering. The final statement means the following: Given a sheaf $\mathcal{G}$ on $T_{pro\text{-}\acute{e}tale}$ and given sections $s_ i \in \Gamma (T_ i, g_{i, small}^{-1}\mathcal{G})$ whose pullbacks to $T_ i \times _ T T_ j$ agree, there is a unique section $s$ of $\mathcal{G}$ over $T$ whose pullback to $T_ i$ agrees with $s_ i$. We will prove this statement when $T$ is affine and the covering is given by a single surjective flat morphism $T' \to T$ of affines and omit the reduction of the general case to this case.

Let $g : T' \to T$ be a surjective flat morphism of affines and let $s' \in g_{small}^{-1}\mathcal{G}(T')$ be a section with $\text{pr}_0^*s' = \text{pr}_1^*s'$ on $T' \times _ T T'$. Choose a surjective weakly étale morphism $W \to T'$ with $W$ affine and weakly contractible, see Lemma 61.13.5. By Lemma 61.16.2 the restriction $s'|_ W$ is an element of $\mathop{\mathrm{colim}}\nolimits _{W \to U} \mathcal{G}(U)$. Choose $\phi : W \to U_0$ and $s_0 \in \mathcal{G}(U_0)$ corresponding to $s'$. Choose a surjective weakly étale morphism $V \to W \times _ T W$ with $V$ affine and weakly contractible. Denote $a, b : V \to W$ the induced morphisms. Since $a^*(s'|_ W) = b^*(s'|_ W)$ and since the category $\{ V \to U, U \in T_{pro\text{-}\acute{e}tale}\}$ is cofiltered (this is clear but see Sites, Lemma 7.14.6 if in doubt), we see that the two morphisms $\phi \circ a , \phi \circ b : V \to U_0$ have to be equal. By the results in Descent, Section 35.13 (especially Descent, Lemma 35.13.7) it follows there is a unique morphism $T \to U_0$ such that $\phi$ is the composition of this morphism with the structure morphism $W \to T$ (small detail omitted). Then we can let $s$ be the pullback of $s_0$ by this morphism. We omit the verification that $s$ pulls back to $s'$ on $T'$. $\square$