hcju/stacksqa huggingface dataset:
The first statement is clear from the construction of the strict henselization as a filtered colimit of \'etale algebras over $S$, or by the characterization of weakly \'etale morphisms of More on Morphisms, Lemma \ref{more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings}. The second statement follows as by Olivier's theorem (More on Algebra, Theorem \ref{more-algebra-theorem-olivier}) the scheme $\Spec(\mathcal{O}_{S, \overline{s}}^{sh})$ is an initial object of the category of pro-\'etale neighbourhoods of $\overline{s}$.
Let $\mathcal{F} = h_U$ be a representable sheaf on $X_\etale$ with $U$ an object of $X_\etale$. In this case $\epsilon^{-1}h_U = h_{u(U)}$ where $u(U)$ is $U$ viewed as an object of $X_\proetale$ (Sites, Lemma \ref{sites-lemma-pullback-representable-sheaf}). Then \begin{align*} (\epsilon^{-1}h_U)(Y) & = h_{u(U)}(Y) \\ & = \Mor_X(Y, U) \\ & = \colim \Mor_X(Y_i, U) \\ & = \colim h_U(Y_i) \end{align*} Here the only nonformal equality is the $3$rd which holds by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation}. Hence the lemma holds for every representable sheaf. Since every sheaf is a coequalizer of a map of coproducts of representable sheaves (Sites, Lemma \ref{sites-lemma-sheaf-coequalizer-representable}) we obtain the result in general.
Suppose that $U$ is a quasi-compact and quasi-separated scheme \'etale over $X$. Then $$ \epsilon_*\epsilon^{-1}\mathcal{F}(U) = \epsilon^{-1}\mathcal{F}(U) = \mathcal{F}(U) $$ the second equality by (a special case of) Lemma \ref{lemma-limit-pullback}. Since every object of $X_\etale$ has a covering by quasi-compact and quasi-separated objects we conclude.
Let $\mathcal{I}$ be an injective abelian sheaf on $X_\etale$. Recall that $R^q\epsilon_*(\epsilon^{-1}\mathcal{I})$ is the sheaf associated to $U \mapsto H^q(U_\proetale, \epsilon^{-1}\mathcal{I})$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. By Lemma \ref{lemma-affine-vanishing} we see that this is zero for $q > 0$ and $U$ affine and \'etale over $X$. Since every object of $X_\etale$ has a covering by affine objects, it follows that $R^q\epsilon_*(\epsilon^{-1}\mathcal{I}) = 0$ for $q > 0$. \medskip\noindent Let $K \in D^+(X_\etale)$. Choose a bounded below complex $\mathcal{I}^\bullet$ of injective abelian sheaves on $X_\etale$ representing $K$. Then $\epsilon^{-1}K$ is represented by $\epsilon^{-1}\mathcal{I}^\bullet$. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) we see that $R\epsilon_*\epsilon^{-1}K$ is represented by $\epsilon_*\epsilon^{-1}\mathcal{I}^\bullet$. By Lemma \ref{lemma-fully-faithful} we conclude that $R\epsilon_*\epsilon^{-1}\mathcal{I}^\bullet = \mathcal{I}^\bullet$ and the proof of (2) is complete. Part (1) is a special case of (2).
Since the functor $\epsilon^{-1}$ is fully faithful by Lemma \ref{lemma-fully-faithful} it is clear that the map $H^1(X_\etale, \mathcal{G}) \to H^1(X_\proetale, \epsilon^{-1}\mathcal{G})$ is injective. To show surjectivity it suffices to show that any $\epsilon^{-1}\mathcal{G}$-torsor $\mathcal{F}$ is \'etale locally trivial. To do this we may assume that $X$ is affine. Thus we reduce to proving surjectivity for $X$ affine. \medskip\noindent Choose a covering $\{U \to X\}$ with (a) $U$ affine, (b) $\mathcal{O}(X) \to \mathcal{O}(U)$ ind-\'etale, and (c) $\mathcal{F}(U)$ nonempty. We can do this by Proposition \ref{proposition-weakly-etale} and the fact that standard pro-\'etale coverings of $X$ are cofinal among all pro-\'etale coverings of $X$ (Lemma \ref{lemma-proetale-affine}). Write $U = \lim U_i$ as a limit of affine schemes \'etale over $X$. Pick $s \in \mathcal{F}(U)$. Let $g \in \epsilon^{-1}\mathcal{G}(U \times_X U)$ be the unique section such that $g \cdot \text{pr}_1^*s = \text{pr}_2^*s$ in $\mathcal{F}(U \times_X U)$. Then $g$ satisfies the cocycle condition $$ \text{pr}_{12}^*g \cdot \text{pr}_{23}^*g = \text{pr}_{13}^*g $$ in $\epsilon^{-1}\mathcal{G}(U \times_X U \times_X U)$. By Lemma \ref{lemma-limit-pullback} we have $$ \epsilon^{-1}\mathcal{G}(U \times_X U) = \colim \mathcal{G}(U_i \times_X U_i) $$ and $$ \epsilon^{-1}\mathcal{G}(U \times_X U \times_X U) = \colim \mathcal{G}(U_i \times_X U_i \times_X U_i) $$ hence we can find an $i$ and an element $g_i \in \mathcal{G}(U_i)$ mapping to $g$ satisfying the cocycle condition. The cocycle $g_i$ then defines a torsor for $\mathcal{G}$ on $X_\etale$ whose pullback is isomorphic to $\mathcal{F}$ by construction. Some details omitted (namely, the relationship between torsors and 1-cocycles which should be added to the chapter on cohomology on sites).
To prove (1) we will prove conditions (1) -- (4) of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Since $\epsilon^{-1}$ is fully faithful (Lemma \ref{lemma-fully-faithful}) and exact, everything is clear except for condition (4). However, if $$ 0 \to \epsilon^{-1}\mathcal{F}_1 \to \mathcal{G} \to \epsilon^{-1}\mathcal{F}_2 \to 0 $$ is a short exact sequence of sheaves of $\Lambda$-modules on $X_\proetale$, then we get $$ 0 \to \epsilon_*\epsilon^{-1}\mathcal{F}_1 \to \epsilon_*\mathcal{G} \to \epsilon_*\epsilon^{-1}\mathcal{F}_2 \to R^1\epsilon_*\epsilon^{-1}\mathcal{F}_1 $$ which by Lemma \ref{lemma-relative-comparison} is the same as a short exact sequence $$ 0 \to \mathcal{F}_1 \to \epsilon_*\mathcal{G} \to \mathcal{F}_2 \to 0 $$ Pulling pack we find that $\mathcal{G} = \epsilon^{-1}\epsilon_*\mathcal{G}$. This proves (1). \medskip\noindent Part (2) follows from part (1) and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-bounded}.
Let $\mathcal{F}$ be a locally constant sheaf of $\Lambda$-modules on $X_\proetale$ of finite presentation. Choose a pro-\'etale covering $\{U_i \to X\}$ such that $\mathcal{F}|_{U_i}$ is constant, say $\mathcal{F}|_{U_i} \cong \underline{M_i}_{U_i}$. Observe that $U_i \times_X U_j$ is empty if $M_i$ is not isomorphic to $M_j$. For each $\Lambda$-module $M$ let $I_M = \{i \in I \mid M_i \cong M\}$. As pro-\'etale coverings are fpqc coverings and by Descent, Lemma \ref{descent-lemma-open-fpqc-covering} we see that $U_M = \bigcup_{i \in I_M} \Im(U_i \to X)$ is an open subset of $X$. Then $X = \coprod U_M$ is a disjoint open covering of $X$. We may replace $X$ by $U_M$ for some $M$ and assume that $M_i = M$ for all $i$. \medskip\noindent Consider the sheaf $\mathcal{I} = \mathit{Isom}(\underline{M}, \mathcal{F})$. This sheaf is a torsor for $\mathcal{G} = \mathit{Isom}(\underline{M}, \underline{M})$. By Modules on Sites, Lemma \ref{sites-modules-lemma-locally-constant} we have $\mathcal{G} = \underline{G}$ where $G = \mathit{Isom}_\Lambda(M, M)$. Since torsors for the \'etale topology and the pro-\'etale topology agree by Lemma \ref{lemma-compare-cohomology-nonabelian} it follows that $\mathcal{I}$ has sections \'etale locally on $X$. Thus $\mathcal{F}$ is \'etale locally a constant sheaf which is what we had to show.
The categories $D_{flc}(X_\etale, \Lambda)$ and $D_{flc}(X_\proetale, \Lambda)$ are strictly full, saturated, triangulated subcategories of $D(X_\etale, \Lambda)$ and $D(X_\proetale, \Lambda)$ by Modules on Sites, Lemma \ref{sites-modules-lemma-kernel-finite-locally-constant} and Derived Categories, Section \ref{derived-section-triangulated-sub} The statement of the lemma follows by combining Lemmas \ref{lemma-compare-derived} and \ref{lemma-compare-locally-constant}.
For assertion (2) this follows from the more general Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-bounded}. The second assertion follows from the fact that the essential image of $\epsilon^{-1}$ is a triangulated subcategory of $D^+(X_\proetale, \Lambda/I^n)$ (Lemma \ref{lemma-compare-derived}), the distinguished triangles $$ K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}} \to K_{n + 1} \to K_n \to K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}}[1] $$ and the isomorphism $$ K \otimes_\Lambda^\mathbf{L} \underline{I^n/I^{n + 1}} = K_1 \otimes_{\Lambda/I}^\mathbf{L} \underline{I^n/I^{n + 1}} $$
Choose generators $f_1, \ldots, f_r$ of $I$. By Algebraic and Formal Geometry, Lemma \ref{algebraization-lemma-derived-completion-koszul} we have $$ K^\wedge =  R\lim (K \otimes_\Lambda^\mathbf{L} \underline{K_n}) $$ where $K_n = K(\Lambda, f_1^n, \ldots, f_r^n)$. In More on Algebra, Lemma \ref{more-algebra-lemma-sequence-Koszul-complexes} we have seen that the pro-systems $\{K_n\}$ and $\{\Lambda/I^n\}$ of $D(\Lambda)$ are isomorphic. Thus the lemma follows.
We have seen (1) and (2) in Algebraic and Formal Geometry, Lemma \ref{algebraization-lemma-pushforward-derived-complete-adjoint}. Part (3) follows from Algebraic and Formal Geometry, Lemma \ref{algebraization-lemma-pushforward-commutes-with-derived-completion}. For (4) let $K$ be derived complete. Then $$ Rf_*K = Rf_*( R\lim K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n}) = R\lim Rf_*(K \otimes^\mathbf{L}_\Lambda \underline{\Lambda/I^n}) $$ the first equality by Lemma \ref{lemma-naive-completion} and the second because $Rf_*$ commutes with $R\lim$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}). This proves (4). To prove (5), by Lemma \ref{lemma-naive-completion} we have $$ Lf_{comp}^*M = R\lim ( Lf^*M \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}) $$ Since $Lf^*$ commutes with derived tensor product by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-tensor-product} and since $Lf^*\underline{\Lambda/I^n} = \underline{\Lambda/I^n}$ we get (5).
Proof in case (2). Set $M_n = M/I^nM$. Then $M = \lim M_n$ as $M$ is assumed $I$-adically complete. Since the action of $G$ is continuous we get continuous actions of $G$ on $M_n$. By \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-equivalence-sheaves-point} this action corresponds to a (locally constant) sheaf $\underline{M_n}$ of $\Lambda/I^n$-modules on $\Spec(k)_\etale$. Pull back to $\Spec(k)_\proetale$ by the comparison morphism $\epsilon$ and take the limit $$ \underline{M}^\wedge = \lim \epsilon^{-1}\underline{M_n} $$ to get the sheaf promised in the lemma. Exactly the same argument as given in the introduction of this section gives the comparison with Tate's continuous Galois cohomology.
Proof of (1). Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$. There is a canonical map $\epsilon^{-1}f_{\etale, *}\mathcal{F} \to f_{\proetale, *}\epsilon^{-1}\mathcal{F}$, see Sites, Section \ref{sites-section-pullback}. To show it is an isomorphism we may work (Zariski) locally on $Y$, hence we may assume $Y$ is affine. In this case every object of $Y_\proetale$ has a covering by objects $V = \lim V_i$ which are limits of affine schemes $V_i$ \'etale over $Y$ (by Proposition \ref{proposition-weakly-etale} for example). Evaluating the map $\epsilon^{-1}f_{\etale, *}\mathcal{F} \to f_{\proetale, *}\epsilon^{-1}\mathcal{F}$ on $V$ we obtain a map $$ \colim \Gamma(X \times_Y V_i, \mathcal{F}) \longrightarrow \Gamma(X \times_Y V, \epsilon^*\mathcal{F}). $$ see Lemma \ref{lemma-limit-pullback} for the left hand side. By Lemma \ref{lemma-limit-pullback} we have $$ \Gamma(X \times_Y V, \epsilon^*\mathcal{F}) = \Gamma(X \times_Y V, \mathcal{F}) $$ Hence the result holds by \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-directed-colimit-cohomology}. \medskip\noindent Proof of (2). Arguing in exactly the same manner as above we see that it suffices to show that $$ \colim H^i_\etale(X \times_Y V_i, \mathcal{F}) \longrightarrow H^i_\etale(X \times_Y V, \mathcal{F}) $$ which follows once more from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-directed-colimit-cohomology}.
The prove this we may work (Zariski) locally on $X$ and assume that $X$ is affine, say $X = \Spec(A)$. Then $Z = \Spec(B)$ for some finite $A$-algebra $B$ of finite presentation. The construction in the proof of Proposition \ref{proposition-find-w-contractible} produces a faithfully flat, ind-\'etale ring map $A \to D$ with $D$ w-contractible. We may check exactness of a sequence of sheaves by evaluating on $U = \Spec(D)$ be such an object. Then $f_{\proetale, *}\mathcal{F}$ evaluated at $U$ is equal to $\mathcal{F}$ evaluated at $V = \Spec(D \otimes_A B)$. Since $D \otimes_A B$ is w-contractible by Lemma \ref{lemma-finite-finitely-presented-over-w-contractible} evaluation at $V$ is exact.
The existence of the adjoint follows immediately from Lemma \ref{lemma-lift-ind-etale} and the definitions. It is clear that $u$ is continuous from the definition of coverings in $X_{app}$. \medskip\noindent Write $X = \Spec(A)$ and $Z = \Spec(A/I)$. Let $V = \Spec(\overline{C})$ be an object of $Z_{app}$ and let $v(V) = \Spec(C)$. We have seen in the statement of Lemma \ref{lemma-lift-ind-etale} that $V$ equals $v(V) \times_X Z = \Spec(C/IC)$. Any $g \in C$ which maps to an invertible element of $C/IC = \overline{C}$ is invertible in $C$. Namely, we have the $A$-algebra maps $C \to C_g \to C/IC$ and by adjointness we obtain an $C$-algebra map $C_g \to C$. Thus every point of $v(V)$ specializes to a point of $V$. \medskip\noindent Suppose that $\{V_i \to V\}$ is a covering in $Z_{app}$. Then $\{v(V_i) \to v(V)\}$ is a finite family of morphisms of $Z_{app}$ such that every point of $V \subset v(V)$ is in the image of one of the maps $v(V_i) \to v(V)$. As the morphisms $v(V_i) \to v(V)$ are flat (since they are weakly \'etale) we conclude that $\{v(V_i) \to v(V)\}$ is jointly surjective. This proves that $v$ sends coverings to coverings. \medskip\noindent Let $V$ be an object of $Z_{app}$ and let $\{U_i \to v(V)\}$ be a covering in $X_{app}$. Then we see that $\{u(U_i) \to u(v(V)) = V\}$ is a covering of $Z_{app}$. By adjointness we obtain morphisms $v(u(U_i)) \to U_i$. Thus the family $\{v(u(U_i)) \to v(V)\}$ refines the given covering and we conclude that $v$ is cocontinuous.
By Lemma \ref{lemma-affine-alternative} we may describe $i_\proetale$ in terms of the morphism of sites $u : X_{app} \to Z_{app}$, $V \mapsto V \times_X Z$. The first statement of the lemma follows from Sites, Lemma \ref{sites-lemma-have-functor-other-way-morphism} (but with the roles of $u$ and $v$ reversed). \medskip\noindent Proof of (1). By the description of $i$ as the morphism of topoi associated to $v$ this holds by the construction, see Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}. \medskip\noindent Proof of (2). Since the functor $v$ sends coverings to coverings by Lemma \ref{lemma-closed-immersion-affines} we see that the presheaf $\mathcal{G} : V \mapsto \mathcal{F}(v(V))$ is a separated presheaf (Sites, Definition \ref{sites-definition-separated}). Hence the sheafification of $\mathcal{G}$ is $\mathcal{G}^+$, see Sites, Theorem \ref{sites-theorem-plus}. Next, let $V$ be a weakly contractible object of $Z_{app}$. Let $\mathcal{V} = \{V_i \to V\}_{i = 1, \ldots, n}$ be any covering in $Z_{app}$. Set $\mathcal{V}' = \{\coprod V_i \to V\}$. Since $v$ commutes with finite disjoint unions (as a left adjoint or by the construction) and since $\mathcal{F}$ sends finite disjoint unions into products, we see that $$ H^0(\mathcal{V}, \mathcal{G}) = H^0(\mathcal{V}', \mathcal{G}) $$ (notation as in Sites, Section \ref{sites-section-sheafification}; compare with \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cech-complex}). Thus we may assume the covering is given by a single morphism, like so $\{V' \to V\}$. Since $V$ is weakly contractible, this covering can be refined by the trivial covering $\{V \to V\}$. It therefore follows that the value of $\mathcal{G}^+ = i^{-1}\mathcal{F}$ on $V$ is simply $\mathcal{F}(v(V))$ and (2) is proved. \medskip\noindent Proof of (3). Every object of $Z_{app}$ has a covering by weakly contractible objects (Lemma \ref{lemma-proetale-enough-w-contractible}). By the above we see that we would have $i^{Sh}_!h_V = h_{v(V)}$ for $V$ weakly contractible if $i^{Sh}_!$ existed. The existence of $i^{Sh}_!$ then follows from Sites, Lemma \ref{sites-lemma-existence-lower-shriek}. \medskip\noindent Proof of (4). Existence of $i_!$ follows in the same way by setting $i_!\mathbf{Z}_V = \mathbf{Z}_{v(V)}$ for $V$ weakly contractible in $Z_{app}$, using similar for direct sums, and applying Homology, Lemma \ref{homology-lemma-partially-defined-adjoint}. Details omitted. \medskip\noindent Proof of (5). Let $V$ be a contractible object of $Z_{app}$. Then $i^{-1}i^{Sh}_!h_V = i^{-1}h_{v(V)} = h_{u(v(V))} = h_V$. (It is a general fact that $i^{-1}h_U = h_{u(U)}$.) Since the sheaves $h_V$ for $V$ contractible generate $\Sh(Z_{app})$ (Sites, Lemma \ref{sites-lemma-sheaf-coequalizer-representable}) we conclude $\text{id} \to i^{-1}i^{Sh}_!$ is an isomorphism. Similarly for the map $\text{id} \to i^{-1}i_!$. Then $(i^{-1}i_*\mathcal{H})(V) = i_*\mathcal{H}(v(V)) = \mathcal{H}(u(v(V))) = \mathcal{H}(V)$ and we find that $i^{-1}i_* \to \text{id}$ is an isomorphism. \medskip\noindent The fully faithfulness statements of (6) now follow from Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
There is an immediate reduction to the case that $X$ is affine. Then $Z$ is affine too. Set $A = \mathcal{O}(X)$ and $B = \mathcal{O}(Z)$. Then the categories of \'etale algebras over $A$ and $B$ are equivalent, see  \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-topological-invariance} and Remark \ref{etale-cohomology-remark-affine-inside-equivalence}. Thus the categories of ind-\'etale algebras over $A$ and $B$ are equivalent. In other words the categories $X_{app}$ and $Z_{app}$ of Lemma \ref{lemma-affine-alternative} are equivalent. We omit the verification that this equivalence sends coverings to coverings and vice versa. Thus the result as Lemma \ref{lemma-affine-alternative} tells us the pro-\'etale topos is the topos of sheaves on $X_{app}$.
Since pullback commutes with restriction, we may replace $X$ by $U$. Thus we may assume that $X$ is affine and weakly contractible and that every point of $X$ specializes to a point of $Z$. By Lemma \ref{lemma-closed-immersion-affines-apply} part (1) it suffices to show that $v(Z) = X$ in this case. Thus we have to show: If $A$ is a w-contractible ring, $I \subset A$ an ideal contained in the Jacobson radical of $A$ and $A \to B \to A/I$ is a factorization with $A \to B$ ind-\'etale, then there is a unique section $B \to A$ compatible with maps to $A/I$. Observe that $B/IB = A/I \times R$ as $A/I$-algebras. After replacing $B$ by a localization we may assume $B/IB = A/I$. Note that $\Spec(B) \to \Spec(A)$ is surjective as the image contains $V(I)$ and hence all closed points and is closed under specialization. Since $A$ is w-contractible there is a section $B \to A$. Since $B/IB = A/I$ this section is compatible with the map to $A/I$. We omit the proof of uniqueness (hint: use that $A$ and $B$ have isomorphic local rings at maximal ideals of $A$).
The question is local on $X$ hence we may assume $X$ is affine. Say $X = \Spec(A)$ and $Z = \Spec(A/I)$. There exist $f_1, \ldots, f_r \in I$ such that $Z = V(f_1, \ldots, f_r)$ set theoretically, see Algebra, Lemma \ref{algebra-lemma-qc-open}. By Lemma \ref{lemma-thickening} we may assume that $Z = \Spec(A/(f_1, \ldots, f_r))$. In this case the functor $i_{\proetale, *}$ is exact by Lemma \ref{lemma-finite}.
Let $\{V_i \to V\}$ be a pro-\'etale covering. The lemma follows if we can refine this covering to a covering where the members have no morphisms into $U$ over $X$ (see construction of $j_!$ in Modules on Sites, Section \ref{sites-modules-section-localize}). First refine the covering to get a finite covering with $V_i$ affine. For each $i$ let $V_i = \Spec(A_i)$ and let $Z_i \subset V_i$ be the inverse image of $Z$. Set $W_i = \Spec(A_{i, Z_i}^\sim)$ with notation as in Lemma \ref{lemma-localization}. Then $\coprod W_i \to V$ is weakly \'etale and the image contains all points of $V_Z$. Hence the image contains all points of $V$ by our assumption on specializations. Thus $\{W_i \to V\}$ is a pro-\'etale covering refining the given one. But each point in $W_i$ specializes to a point lying over $Z$, hence there are no morphisms $W_i \to U$ over $X$.
We obtain the maps by the adjointness properties of the functors involved. It suffices to show that $X_\proetale$ has enough objects (Sites, Definition \ref{sites-definition-w-contractible}) on which the sequence evaluates to a short exact sequence. Let $V = \Spec(A)$ be an affine object of $X_\proetale$ such that $A$ is w-contractible (there are enough objects of this type). Then $V \times_X Z$ is cut out by an ideal $I \subset A$. The assumption that $j$ is quasi-compact implies there exist $f_1, \ldots, f_r \in I$ such that $V(I) = V(f_1, \ldots, f_r)$. We obtain a faithfully flat, ind-Zariski ring map $$ A \longrightarrow A_{f_1} \times \ldots \times A_{f_r} \times A_{V(I)}^\sim $$ with $A_{V(I)}^\sim$ as in Lemma \ref{lemma-localization}. Since $V_i = \Spec(A_{f_i}) \to X$ factors through $U$ we have $$ j_!j^{-1}\mathcal{F}(V_i) = \mathcal{F}(V_i) \quad\text{and}\quad i_*i^{-1}\mathcal{F}(V_i) = 0 $$ On the other hand, for the scheme $V^\sim = \Spec(A_{V(I)}^\sim)$ we have $$ j_!j^{-1}\mathcal{F}(V^\sim) = 0 \quad\text{and}\quad \mathcal{F}(V^\sim) = i_*i^{-1}\mathcal{F}(V^\sim) $$ the first equality by Lemma \ref{lemma-jshriek-zero} and the second by Lemmas \ref{lemma-compute-i-star} and \ref{lemma-localization-w-contractible}. Thus the sequence evaluates to an exact sequence on $\Spec(A_{f_1} \times \ldots \times A_{f_r} \times A_{V(I)}^\sim)$ and the lemma is proved.
Since $j_!$ is exact it suffices to show that $j_!$ commutes with products. The question is local on $X$, hence we may assume $X$ affine. Let $\mathcal{G}$ be an abelian sheaf on $U_\proetale$. We have $j^{-1}j_*\mathcal{G} = \mathcal{G}$. Hence applying the exact sequence of Lemma \ref{lemma-ses-associated-to-open} we get $$ 0 \to j_!\mathcal{G} \to j_*\mathcal{G} \to i_*i^{-1}j_*\mathcal{G} \to 0 $$ where $i : Z \to X$ is the inclusion of the reduced induced scheme structure on the complement $Z = X \setminus U$. The functors $j_*$ and $i_*$ commute with products as right adjoints. The functor $i^{-1}$ commutes with products by Lemma \ref{lemma-closed-immersion}. Hence $j_!$ does because on the pro-\'etale site products are exact (Cohomology on Sites, Proposition \ref{sites-cohomology-proposition-enough-weakly-contractibles}).