hcju/stacksqa huggingface dataset:
Denote the functor $u : T_\proetale \to (\Sch/S)_\proetale$. In other words, given a weakly \'etale morphism $j : U \to T$ corresponding to an object of $T_\proetale$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma \ref{lemma-fibre-products-proetale}. Moreover, $T_\proetale$ has equalizers and $u$ commutes with them by Lemma \ref{lemma-fibre-products-proetale}. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers}.
The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers (details omitted; compare with proof of Lemma \ref{lemma-put-in-T}). Hence Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers} apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/S$ we have $\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)$ as desired. Thus we may apply Sites, Lemmas \ref{sites-lemma-have-functor-other-way} and \ref{sites-lemma-have-functor-other-way-morphism} to get the formula for $f_{big, *}$.
The equality $i_f = f_{big} \circ i_T$ follows from the equality $i_f^{-1} = i_T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1). \medskip\noindent The functor $u : S_\proetale \to T_\proetale$, $u(U \to S) = (U \times_S T \to T)$ transforms coverings into coverings and commutes with fibre products, see Lemmas \ref{lemma-proetale} and \ref{lemma-fibre-products-proetale}. Moreover, both $S_\proetale$, $T_\proetale$ have final objects, namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by Sites, Proposition \ref{sites-proposition-get-morphism} the functor $u$ corresponds to a morphism of sites $T_\proetale \to S_\proetale$. This in turn gives rise to the morphism of topoi, see Sites, Lemma \ref{sites-lemma-morphism-sites-topoi}. The description of the pushforward is clear from these references. \medskip\noindent Part (3) follows because $\pi_S$ and $\pi_T$ are given by the inclusion functors and $f_{small}$ and $f_{big}$ by the base change functors $U \mapsto U \times_S T$. \medskip\noindent Statement (4) follows from (3) by precomposing with $i_T$.
Since the diagram is cartesian, we have for $U'/S'$ that $U' \times_{S'} T' = U' \times_S T$. Hence both $i_g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\Sch/T)_\proetale$ to the sheaf $U' \mapsto \mathcal{F}(U' \times_{S'} T')$ on $S'_\proetale$ (use Lemmas \ref{lemma-put-in-T} and \ref{lemma-morphism-big}). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma \ref{sites-lemma-localize-morphism}.
By Lemma \ref{lemma-alternative} we may replace $S_\proetale$ by $S_{affine, \proetale}$. The lemma follows from Sites, Lemma \ref{sites-lemma-equivalence} by checking that the inclusion functor $S_{app} \to S_{affine, \proetale}$ is a special cocontinuous functor, see Sites, Definition \ref{sites-definition-special-cocontinuous-functor}. The conditions of Sites, Lemma \ref{sites-lemma-equivalence} follow immediately from the definition and the facts (a) any object $U$ of $S_{affine, \proetale}$ has a covering $\{V \to U\}$ with $V$ ind-\'etale over $X$ (Proposition \ref{proposition-weakly-etale}) and (b) the functor $u$ is fully faithful.
Assume $A$ is w-contractible. By Lemma \ref{lemma-proetale-affine} it suffices to prove we can refine every standard pro-\'etale covering $\{f_i : T_i \to T\}_{i = 1, \ldots, n}$ by a Zariski covering of $T$. The morphism $\coprod T_i \to T$ is a surjective weakly \'etale morphism of affine schemes. Hence by Definition \ref{definition-w-contractible} there exists a morphism $\sigma : T \to \coprod T_i$ over $T$. Then the Zariski covering $T = \coprod \sigma^{-1}(T_i)$ refines $\{f_i : T_i \to T\}$. \medskip\noindent Conversely, assume (2). If $A \to B$ is faithfully flat and weakly \'etale, then $\{\Spec(B) \to T\}$ is a pro-\'etale covering. Hence there exists a Zariski covering $T = \coprod U_i$ and morphisms $U_i \to \Spec(B)$ over $T$. Since $T = \coprod U_i$ we obtain $T \to \Spec(B)$, i.e., an $A$-algebra map $B \to A$. This means $A$ is w-contractible.
We have seen the equivalence of (1) and (3) in Lemma \ref{lemma-w-contractible-proetale-cover}. \medskip\noindent Assume (3) and let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves on $\Sch_\proetale$. Let $s \in \mathcal{G}(T)$. To prove (2) we will show that $s$ is in the image of $\mathcal{F}(T) \to \mathcal{G}(T)$. We can find a covering $\{T_i \to T\}$ of $\Sch_\proetale$ such that $s$ lifts to a section of $\mathcal{F}$ over $T_i$ (Sites, Definition \ref{sites-definition-sheaves-injective-surjective}). By (3) we may assume we have a finite covering $T = \coprod_{j = 1, \ldots, m} U_j$ by open and closed subsets and we have $t_j \in \mathcal{F}(U_j)$ mapping to $s|_{U_j}$. Since Zariski coverings are coverings in $\Sch_\proetale$ (Lemma \ref{lemma-etale-proetale}) we conclude that $\mathcal{F}(T) = \prod \mathcal{F}(U_j)$. Thus $t = (t_1, \ldots, t_m) \in \mathcal{F}(T)$ is a section mapping to $s$. \medskip\noindent Assume (2). Let $A \to D$ be as in Proposition \ref{proposition-find-w-contractible}. Then $\{V \to T\}$ is a covering of $\Sch_\proetale$. (Note that $V = \Spec(D)$ is an object of $\Sch_\proetale$ by Remark \ref{remark-size-w-contractible} combined with our choice of the function $Bound$ in Definition \ref{definition-big-proetale-site} and the computation of the size of affine schemes in Sets, Lemma \ref{sets-lemma-bound-size}.) Since the topology on $\Sch_\proetale$ is subcanonical (Lemma \ref{lemma-proetale-subcanonical}) we see that $h_V \to h_T$ is a surjective map of sheaves (Sites, Lemma \ref{sites-lemma-covering-surjective-after-sheafification}). Since $T$ is assumed weakly contractible, we see that there is an element $f \in h_V(T) = \Mor(T, V)$ whose image in $h_T(T)$ is $\text{id}_T$. Thus $A \to D$ has a section $\sigma : D \to A$. Now if $A \to B$ is faithfully flat and weakly \'etale, then $D \to D \otimes_A B$ has the same properties, hence there is a section $D \otimes_A B \to D$ and combined with $\sigma$ we get a section $B \to D \otimes_A B \to D \to A$ of $A \to B$. Thus $A$ is w-contractible and (1) holds.
For those readers who do not care about set-theoretical issues this lemma is a trivial consequence of Lemma \ref{lemma-w-contractible-is-weakly-contractible} and Proposition \ref{proposition-find-w-contractible}. Here are the details. Choose an affine open covering $T = \bigcup U_i$. Write $U_i = \Spec(A_i)$. Choose faithfully flat, ind-\'etale ring maps $A_i \to D_i$ such that $D_i$ is w-contractible as in Proposition \ref{proposition-find-w-contractible}. The family of morphisms $\{\Spec(D_i) \to T\}$ is a pro-\'etale covering. If we can show that $\Spec(D_i)$ is isomorphic to an object, say $T_i$, of $\Sch_\proetale$, then $\{T_i \to T\}$ will be combinatorially equivalent to a covering of $\Sch_\proetale$ by the construction of $\Sch_\proetale$ in Definition \ref{definition-big-proetale-site} and more precisely the application of Sets, Lemma \ref{sets-lemma-coverings-site} in the last step. To prove $\Spec(D_i)$ is isomorphic to an object of $\Sch_\proetale$, it suffices to prove that $|D_i| \leq Bound(\text{size}(T))$ by the construction of $\Sch_\proetale$ in Definition \ref{definition-big-proetale-site} and more precisely the application of Sets, Lemma \ref{sets-lemma-construct-category} in step (3). Since $|A_i| \leq \text{size}(U_i) \leq \text{size}(T)$ by Sets, Lemmas \ref{sets-lemma-bound-affine} and \ref{sets-lemma-bound-finite-type} we get $|D_i| \leq \kappa^{2^{2^{2^\kappa}}}$ where $\kappa = \text{size}(T)$ by Remark \ref{remark-size-w-contractible}. Thus by our choice of the function $Bound$ in Definition \ref{definition-big-proetale-site} we win.
Suppose that $V = \coprod_{j \in J} V_j$ is an object of $(\Sch/S)_\proetale$ which is the disjoint union of weakly contractible objects $V_j$. Since a disjoint union decomposition is a pro-\'etale covering we see that $\mathcal{F}(V) = \prod_{j \in J} \mathcal{F}(V_j)$ for any pro-\'etale sheaf $\mathcal{F}$. Let $\mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves of sets. Since $V_j$ is weakly contractible, the map $\mathcal{F}(V_j) \to \mathcal{G}(V_j)$ is surjective, see Sites, Definition \ref{sites-definition-w-contractible}. Thus $\mathcal{F}(V) \to \mathcal{G}(V)$ is surjective as a product of surjective maps of sets and we conclude that $V$ is weakly contractible. \medskip\noindent Choose a covering $\{U_i \to U\}_{i \in I}$ with $U_i$ affine and weakly contractible as in Lemma \ref{lemma-get-many-weakly-contractible}. Take $V = \coprod_{i \in I} U_i$ (there is a set theoretic issue here which we will address below). Then $\{V \to U\}$ is the desired pro-\'etale covering by a weakly contractible object (to check it is a covering use Lemma \ref{lemma-recognize-proetale-covering}). If $U$ is quasi-compact, then it follows immediately from Lemma \ref{lemma-recognize-proetale-covering} that we can choose a finite subset $I' \subset I$ such that $\{U_i \to U\}_{i \in I'}$ is still a covering and then $\{\coprod_{i \in I'} U_i \to U\}$ is the desired covering by an affine and weakly contractible object. \medskip\noindent In this paragraph, which we urge the reader to skip, we address set theoretic problems. In order to know that the disjoint union lies in our partial universe, we need to bound the cardinality of the index set $I$. It is seen immediately from the construction of the covering $\{U_i \to U\}_{i \in I}$ in the proof of Lemma \ref{lemma-get-many-weakly-contractible} that $|I| \leq \text{size}(U)$ where the size of a scheme is as defined in Sets, Section \ref{sets-section-categories-schemes}. Moreover, for each $i$ we have $\text{size}(U_i) \leq Bound(\text{size}(U))$; this follows for the bound of the cardinality of $\Gamma(U_i, \mathcal{O}_{U_i})$ in the proof of Lemma \ref{lemma-get-many-weakly-contractible} and Sets, Lemma \ref{sets-lemma-bound-affine}. Thus $\text{size}(\coprod_{i \in I} U_i)) \leq Bound(\text{size}(U))$ by Sets, Lemma \ref{sets-lemma-bound-size}. Hence by construction of the big pro-\'etale site through Sets, Lemma \ref{sets-lemma-construct-category} we see that $\coprod_{i \in I} U_i$ is isomorphic to an object of our site and the proof is complete.
Let $\mathcal{B} \subset \Ob(X_\proetale)$ be the set of weakly contractible objects of $X_\proetale$. Every object $T$ of $X_\proetale$ has a covering $\{T_i \to T\}_{i \in I}$ with $I$ finite and $T_i \in \mathcal{B}$ by Lemma \ref{lemma-weakly-contractible-cover}. By Hypercoverings, Lemma \ref{hypercovering-lemma-w-contractible} we get a hypercovering $K$ of $U$ such that $K_n = \{U_{n, i}\}_{i \in I_n}$ with $I_n$ finite and $U_{n, i}$ weakly contractible. Then we can replace $K$ by the hypercovering of $U$ given by $\{U_n\}$ in degree $n$ where $U_n = \coprod_{i \in I_n} U_{n, i}$ This is allowed by Hypercoverings, Remark \ref{hypercovering-remark-take-unions-hypercovering-X}. \medskip\noindent Let $X_{qcqs, \proetale} \subset X_\proetale$ be the full subcategory consisting of quasi-compact and quasi-separated objects. A covering of $X_{qcqs, \proetale}$ will be a finite pro-\'etale covering. Then $X_{qcqs, \proetale}$ is a site, has fibre products, and the inclusion functor $X_{qcqs, \proetale} \to X_\proetale$ is continuous and commutes with fibre products. In particular, if $K$ is a hypercovering of an object $U$ in $X_{qcqs, \proetale}$ then $K$ is a hypercovering of $U$ in $X_\proetale$ by Hypercoverings, Lemma \ref{hypercovering-lemma-hypercovering-continuous-functor}. Let $\mathcal{B} \subset \Ob(X_{qcqs, \proetale})$ be the set of affine and weakly contractible objects. By  Lemma \ref{lemma-get-many-weakly-contractible} and the fact that finite unions of affines are affine, for every object $U$ of $X_{qcqs, \proetale}$ there exists a covering $\{V \to U\}$ of $X_{qcqs, \proetale}$ with $V \in \mathcal{B}$. By Hypercoverings, Lemma \ref{hypercovering-lemma-w-contractible} we get a hypercovering $K$ of $U$ such that $K_n = \{U_{n, i}\}_{i \in I_n}$ with $I_n$ finite and $U_{n, i}$ affine and weakly contractible. Then we can replace $K$ by the hypercovering of $U$ given by $\{U_n\}$ in degree $n$ where $U_n = \coprod_{i \in I_n} U_{n, i}$. This is allowed by Hypercoverings, Remark \ref{hypercovering-remark-take-unions-hypercovering-X}.
If $\mathcal{E}$ is an abelian sheaf on $X_\proetale$, then the spectral sequence of Hypercoverings, Lemma \ref{hypercovering-lemma-cech-spectral-sequence} implies that $$ R\Gamma(X_\proetale, \mathcal{E}) = s(\mathcal{E}(K)) $$ because the higher cohomology groups of any sheaf over $U_n$ vanish, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-w-contractible}. \medskip\noindent If $\mathcal{E}^\bullet$ is bounded below, then we can choose an injective resolution $\mathcal{E}^\bullet \to \mathcal{I}^\bullet$ and consider the map of complexes $$ \text{Tot}(s(\mathcal{E}^\bullet(K))) \longrightarrow \text{Tot}(s(\mathcal{I}^\bullet(K))) $$ For every $n$ the map $\mathcal{E}^\bullet(U_n) \to \mathcal{I}^\bullet(U_n)$ is a quasi-isomorphism because taking sections over $U_n$ is exact. Hence the displayed map is a quasi-isomorphism by one of the spectral sequences of Homology, Lemma \ref{homology-lemma-first-quadrant-ss}. Using the result of the first paragraph we see that for every $p$ the complex $s(\mathcal{I}^p(K))$ is acyclic in degrees $n > 0$ and computes $\mathcal{I}^p(U)$ in degree $0$. Thus the other spectral sequence of Homology, Lemma \ref{homology-lemma-first-quadrant-ss} shows $\text{Tot}(s(\mathcal{I}^\bullet(K)))$ computes $R\Gamma(U, E) = \mathcal{I}^\bullet(U)$.
The statement means the following: Suppose we have a family of objects $E_i$ of $D^+(X_\proetale)$ such that $\bigoplus E_i$ is an object of $D^+(X_\proetale)$. Then $R\Gamma(X, \bigoplus E_i) = \bigoplus R\Gamma(X, E_i)$. To see this choose a hypercovering $K$ of $X$ with $K_n = \{U_n \to X\}$ where $U_n$ is an affine and weakly contractible scheme, see Lemma \ref{lemma-w-contractible-hypercovering}. Let $N$ be an integer such that $H^p(E_i) = 0$ for $p < N$. Choose a complex of abelian sheaves $\mathcal{E}_i^\bullet$ representing $E_i$ with $\mathcal{E}_i^p = 0$ for $p < N$. The termwise direct sum $\bigoplus \mathcal{E}_i^\bullet$ represents $\bigoplus E_i$ in $D(X_\proetale)$, see Injectives, Lemma \ref{injectives-lemma-derived-products}. By Lemma \ref{lemma-compute-cohomology} we have $$ R\Gamma(X, \bigoplus E_i) = \text{Tot}(s((\bigoplus \mathcal{E}^\bullet_i)(K))) $$ and $$ R\Gamma(X, E_i) = \text{Tot}(s(\mathcal{E}^\bullet_i(K))) $$ Since each $U_n$ is quasi-compact we see that $$ \text{Tot}(s((\bigoplus \mathcal{E}^\bullet_i)(K))) = \bigoplus \text{Tot}(s(\mathcal{E}^\bullet_i(K))) $$ by Modules on Sites, Lemma \ref{sites-modules-lemma-sections-over-quasi-compact}. The statement on homotopy colimits is a formal consequence of the fact that $R\Gamma$ is an exact functor of triangulated categories and the fact (just proved) that it commutes with direct sums.
Proof of (3). Let $U$ be an affine object of $S_\proetale$ which is weakly contractible. Then $j_{U!}\mathcal{A}_U$ is a compact object of the derived category $D(S_\proetale, \mathcal{A})$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-quasi-compact-weakly-contractible-compact}. Choose a set $I$ and for each $i \in I$ an affine weakly contractible object $U_i$ of $S_\proetale$ such that every affine weakly contractible object of $S_\proetale$ is isomorphic to one of the $U_i$. This is possible because $\Ob(S_\proetale)$ is a set. To finish the proof of (3) it suffices to show that $\bigoplus j_{U_i, !}\mathcal{A}_{U_i}$ is a generator of $D(S_\proetale, \mathcal{A})$, see Derived Categories, Definition \ref{derived-definition-generators}. To see this, let $K$ be a nonzero object of $D(S_\proetale, \mathcal{A})$. Then there exists an object $T$ of our site $S_\proetale$ and a nonzero element $\xi$ of $H^n(K)(T)$. In other words, $\xi$ is a nonzero section of the $n$th cohomology sheaf of $K$. We may assume $K$ is represented by a complex $\mathcal{K}^\bullet$ of sheaves of $\mathcal{A}$-modules and $\xi$ is the class of a section $s \in \mathcal{K}^n(T)$ with $\text{d}(s) = 0$. Namely, $\xi$ is locally represented as the class of a section (so you get the result after replacing $T$ by a member of a covering of $T$). Next, we choose a covering $\{T_j \to T\}_{j \in J}$ as in Lemma \ref{lemma-get-many-weakly-contractible}. Since $H^n(K)$ is a sheaf, we see that for some $j$ the restriction $\xi|_{T_j}$ remains nonzero. Thus $s|_{T_j}$ defines a nonzero map $j_{T_j, !}\mathcal{A}_{T_j} \to K$ in $D(S_\proetale, \mathcal{A})$. Since $T_j \cong U_i$ for some $i \in I$ we conclude. \medskip\noindent The exact same argument works for the big pro-\'etale site of $S$.
Recall that $\mathcal{F}^\#$ is equal to $(\mathcal{F}^+)^+$, see Sites, Theorem \ref{sites-theorem-plus}, where $\mathcal{F}^+$ is the presheaf which sends an object $U$ of $X_\proetale$ to $\colim H^0(\mathcal{U}, \mathcal{F})$ where the colimit is over all pro-\'etale 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-\'etale covering $\mathcal{U} = \{f_j : V_j \to U\}$ of $U$ we obtain pro-\'etale coverings $\mathcal{U}_i = \{f_j^{-1}(U_i) \to U_i\}$ and we clearly have $$ H^0(\mathcal{U}, \mathcal{F}) = H^0(\mathcal{U}_1, \mathcal{F}) \times H^0(\mathcal{U}_2, \mathcal{F}) $$ 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-\'etale covering of $W$ can be refined by a finite disjoint union decomposition $W = W_1 \amalg \ldots W_n$ by Lemma \ref{lemma-w-contractible-is-weakly-contractible}. 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).
Recall that $f_{small}^{-1}\mathcal{F}$ is the sheaf associated to the presheaf $$ u_p\mathcal{F} : U \mapsto \colim_{U \to V} \mathcal{F}(V) $$ on $X_\etale$, see Sites, Sections \ref{sites-section-morphism-sites} and \ref{sites-section-continuous-functors}; we've surpressed from the notation that the colimit is over the opposite of the category $\{U \to V, V \in Y_\proetale\}$. By Lemma \ref{lemma-presheaf-value-weakly-contractible} 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 $$ \{U_1 \to V_1\} \times \{U_2 \to V_2\} \longrightarrow \{U \to V\},\quad (U_1 \to V_1, U_2 \to V_2) \longmapsto (U \to V_1 \amalg V_2) $$ which is initial (Categories, Definition \ref{categories-definition-initial}). Hence the corresponding functor on opposite categories is cofinal and by Categories, Lemma \ref{categories-lemma-cofinal} 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.
Observe that we have a morphism $i_f : \Sh(T_\proetale) \to \Sh(\Sch/S)_\proetale)$ such that $\pi_S \circ i_f = f_{small}$ as morphisms $T_\proetale \to S_\proetale$, see Lemma \ref{lemma-put-in-T}. Since pullback is transitive we see that $i_f^{-1} \pi_S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$ as desired. \medskip\noindent 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 $T_\proetale$ 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. \medskip\noindent 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 \'etale morphism $W \to T'$ with $W$ affine and weakly contractible, see Lemma \ref{lemma-weakly-contractible-cover}. By Lemma \ref{lemma-small-pullback-weakly-contractible} the restriction $s'|_W$ is an element of $\colim_{W \to U} \mathcal{G}(U)$. Choose $\phi : W \to U_0$ and $s_0 \in \mathcal{G}(U_0)$ corresponding to $s'$. Choose a surjective weakly \'etale 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 $\{V \to U, U \in T_\proetale\}$ is cofiltered (this is clear but see Sites, Lemma \ref{sites-lemma-directed-morphism} 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 \ref{descent-section-fpqc-universal-effective-epimorphisms} (especially Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}) 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 $T'$.
Proof of (1)(a) and (2)(a): $i_f^{-1}$ is a right adjoint of an exact functor $i_{f, !}$. Namely, recall that $i_f$ corresponds to a cocontinuous functor $u : T_\proetale \to (\Sch/S)_\proetale$ which is continuous and commutes with fibre products and equalizers, see Lemma \ref{lemma-put-in-T} and its proof. Hence we obtain $i_{f, !}$ by Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}. It is shown in Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek} that it is exact. Then we conclude (1)(a) and (2)(a) hold by Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} and Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. \medskip\noindent Parts (1)(b) and (2)(b) are special cases of (1)(a) and (2)(a) as $i_S = i_{\text{id}_S}$.
The first equality follows from Lemma \ref{lemma-compare-injectives} and (\ref{equation-compare-big-small}) on choosing a K-injective complex of abelian sheaves representing $K$. The second equality follows from Lemma \ref{lemma-compare-injectives} and Lemma \ref{lemma-morphism-big-small-cartesian-diagram} on choosing a K-injective complex of abelian sheaves representing $K$. The third equality follows similarly from Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and \ref{sites-cohomology-lemma-restrict-K-injective-to-open} and Lemma \ref{lemma-morphism-big-small-cartesian-diagram} on choosing a K-injective complex of abelian sheaves representing $K$.
The big pro-\'etale topos of $S$ is equivalent to the topos defined by $(\textit{Aff}/S)_\proetale$, see Lemma \ref{lemma-affine-big-site-proetale}. The topos of sheaves on $S_\proetale$ is equivalent to the topos associated to $S_{affine, \proetale}$, see Lemma \ref{lemma-alternative}. The result for the sites $(\textit{Aff}/S)_\proetale$ and $S_{affine, \proetale}$ follows immediately from Deligne's result Sites, Lemma \ref{sites-lemma-criterion-points}. The case $\Sch_\proetale$ is handled because it is equal to $(\Sch/\Spec(\mathbf{Z}))_\proetale$.
The proof is identitical to the proof of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cofinal-etale} but using the corresponding facts about weakly \'etale morphisms proven in More on Morphisms, Lemmas \ref{more-morphisms-lemma-composition-weakly-etale}, \ref{more-morphisms-lemma-base-change-weakly-etale}, and \ref{more-morphisms-lemma-weakly-etale-permanence}.