hcju/stacksqa huggingface dataset:
Observe that the corresponding map $$ B(Y') \longrightarrow B(Y) \times_{B(Z)} B(Z') $$ is a bijection, see Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}. Thus using the commutative diagram $$ \xymatrix{ \Complexesstack_{X/B, Y'} \ar[r] \ar[d] & \Complexesstack_{X/B, Y} \times_{\Complexesstack_{X/B, Z}} \Complexesstack_{X/B, Z'} \ar[d] \\ B(Y') \ar[r] & B(Y) \times_{B(Z)} B(Z') } $$ we see that we may assume that $Y'$ is a scheme over $B'$. By Remark \ref{remark-complexes-base-change} we may replace $B$ by $Y'$ and $X$ by $X \times_B Y'$. Thus we may assume $B = Y'$. \medskip\noindent Assume $B = Y'$. We first prove fully faithfulness of our functor. To do this, let $\xi_1, \xi_2$ be two objects of $\Complexesstack_{X/B}$ over $Y'$. Then we have to show that $$ \mathit{Isom}(\xi_1, \xi_2)(Y') \longrightarrow \mathit{Isom}(\xi_1, \xi_2)(Y) \times_{\mathit{Isom}(\xi_1, \xi_2)(Z)} \mathit{Isom}(\xi_1, \xi_2)(Z') $$ is bijective. However, we already know that $\mathit{Isom}(\xi_1, \xi_2)$ is an algebraic space over $B = Y'$. Thus this bijectivity follows from Artin's Axioms, Lemma \ref{artin-lemma-pushout} (or the aforementioned Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}). \medskip\noindent Essential surjectivity. Let $(E_Y, E_{Z'}, \alpha)$ be a triple, where $E_Y \in D(\mathcal{O}_Y)$ and $E_{Z'} \in D(\mathcal{O}_{X_{Z'}})$ are objects such that $(Y, Y \to B, E_Y)$ is an object of $\Complexesstack_{X/B}$ over $Y$, such that $(Z', Z' \to B, E_{Z'})$ is an object of $\Complexesstack_{X/B}$ over $Z'$, and $\alpha : L(X_Z \to X_Y)^*E_Y \to L(X_Z \to X_{Z'})^*E_{Z'}$ is an isomorphism in $D(\mathcal{O}_{Z'})$. That is to say $$ ((Y, Y \to B, E_Y), (Z', Z' \to B, E_{Z'}), \alpha) $$ is an object of the target of the arrow of our lemma. Observe that the diagram $$ \xymatrix{ X_Z \ar[r] \ar[d] & X_{Z'} \ar[d] \\ X_Y \ar[r] & X_{Y'} } $$ is a pushout with $X_Z \to X_Y$ affine and $X_Z \to X_{Z'}$ a thickening (see Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat}). Hence by Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-pushout-along-thickening-derived} we find an object $E_{Y'} \in D(\mathcal{O}_{X_{Y'}})$ together with isomorphisms $L(X_Y \to X_{Y'})^*E_{Y'} \to E_Y$ and $L(X_{Z'} \to X_{Y'})^*E_{Y'} \to E_Z$ compatible with $\alpha$. Clearly, if we show that $E_{Y'}$ is $Y'$-perfect, then we are done, because property (2) of Lemma \ref{lemma-complexes} is a property on points (and $Y$ and $Y'$ have the same points). This follows from More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-thickening-relatively-perfect}.
Observe that by Lemma \ref{lemma-complexes-RS-star} our stack in groupoids $\mathcal{X}$ satisfies property (RS*) defined in Artin's Axioms, Section \ref{artin-section-RS-star}. In particular $\mathcal{X}$ satisfies (RS). Hence all associated predeformation categories are deformation categories (Artin's Axioms, Lemma \ref{artin-lemma-deformation-category}) and the statement makes sense. \medskip\noindent In this paragraph we show that we can reduce to the case $B = \Spec(k)$. Set $X_0 = \Spec(k) \times_{g_0, B} X$ and denote $\mathcal{X}_0 = \Complexesstack_{X_0/k}$. In Remark \ref{remark-complexes-base-change} we have seen that $\mathcal{X}_0$ is the $2$-fibre product of $\mathcal{X}$ with $\Spec(k)$ over $B$ as categories fibred in groupoids over $(\Sch/S)_{fppf}$. Thus by Artin's Axioms, Lemma \ref{artin-lemma-fibre-product-tangent-spaces} we reduce to proving that $B$, $\Spec(k)$, and $\mathcal{X}_0$ have finite dimensional tangent spaces and infinitesimal automorphism spaces. The tangent space of $B$ and $\Spec(k)$ are finite dimensional by Artin's Axioms, Lemma \ref{artin-lemma-finite-dimension} and of course these have vanishing $\text{Inf}$. Thus it suffices to deal with $\mathcal{X}_0$. \medskip\noindent Let $k[\epsilon]$ be the dual numbers over $k$. Let $\Spec(k[\epsilon]) \to B$ be the composition of $g_0 : \Spec(k) \to B$ and the morphism $\Spec(k[\epsilon]) \to \Spec(k)$ coming from the inclusion $k \to k[\epsilon]$. Set $X_0 = \Spec(k) \times_B X$ and $X_\epsilon = \Spec(k[\epsilon]) \times_B X$. Observe that $X_\epsilon$ is a first order thickening of $X_0$ flat over the first order thickening $\Spec(k) \to \Spec(k[\epsilon])$. Observe that $X_0$ and $X_\epsilon$ give rise to canonically equivalent small \'etale topoi, see More on Morphisms of Spaces, Section \ref{spaces-more-morphisms-section-thickenings}. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-thickening-relatively-perfect} we see that $T\mathcal{F}_{\mathcal{X}_0, k, x_0}$ is the set of isomorphism classes of lifts of $E_0$ to $X_\epsilon$ in the sense of Deformation Theory, Lemma \ref{defos-lemma-first-order-defos-complex}. We conclude that $$ T\mathcal{F}_{\mathcal{X}_0, k, x_0} = \Ext^1_{\mathcal{O}_{X_0}}(E_0, E_0) $$ Here we have used the identification $\epsilon k[\epsilon] \cong k$ of $k[\epsilon]$-modules. Using Deformation Theory, Lemma \ref{defos-lemma-first-order-defos-complex} once more we see that there is a surjection $$ \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) \leftarrow \Ext^0_{\mathcal{O}_{X_0}}(E_0, E_0) $$ of $k$-vector spaces. As $E_0$ is pseudo-coherent it lies in $D^-_{\textit{Coh}}(\mathcal{O}_{X_0})$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-identify-pseudo-coherent-noetherian}. Since $E_0$ locally has finite tor dimension and $X_0$ is quasi-compact we see $E_0 \in D^b_{\textit{Coh}}(\mathcal{O}_{X_0})$. Thus the $\Ext$s above are finite dimensional $k$-vector spaces by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-ext-finite}.
Let $(R_n)$ be an inverse system of $S$-algebras with surjective transition maps whose kernels are locally nilpotent. Set $R = \lim R_n$. Let $(\xi_n)$ be a system of objects of $\Complexesstack_{X/B}$ lying over $(\Spec(R_n))$. We have to show $(\xi_n)$ is effective, i.e., there exists an object $\xi$ of $\Complexesstack_{X/B}$ lying over $\Spec(R)$. \medskip\noindent Write $X_R = \Spec(R) \times_S X$ and $X_n = \Spec(R_n) \times_S X$. Of course $X_n$ is the base change of $X_R$ by $R \to R_n$. Since $S = B$, we see that $\xi_n$ corresponds simply to an $R_n$-perfect object $E_n \in D(\mathcal{O}_{X_n})$ satisfying condition (2) of Lemma \ref{lemma-complexes}. In particular $E_n$ is pseudo-coherent. The isomorphisms $\xi_{n + 1}|_{\Spec(R_n)} \cong \xi_n$ correspond to isomorphisms $L(X_n \to X_{n + 1})^*E_{n + 1} \to E_n$. Therefore by Flatness on Spaces, Theorem \ref{spaces-flat-theorem-existence-derived} we find a pseudo-coherent object $E$ of $D(\mathcal{O}_{X_R})$ with $E_n$ equal to the derived pullback of $E$ for all $n$ compatible with the transition isomorphisms. \medskip\noindent Observe that $(R, \Ker(R \to R_1))$ is a henselian pair, see More on Algebra, Lemma \ref{more-algebra-lemma-limit-henselian}. In particular, $\Ker(R \to R_1)$ is contained in the Jacobson radical of $R$. Then we may apply More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-henselian-relatively-perfect} to see that $E$ is $R$-perfect. \medskip\noindent Finally, we have to check condition (2) of Lemma \ref{lemma-complexes}. By Lemma \ref{lemma-complexes-open-neg-exts-vanishing} the set of points $t$ of $\Spec(R)$ where the negative self-exts of $E_t$ vanish is an open. Since this condition is true in $V(\Ker(R \to R_1))$ and since $\Ker(R \to R_1)$ is contained in the Jacobson radical of $R$ we conclude it holds for all points.
By Lemma \ref{lemma-hom-sheaf} the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property for fppf coverings. This mean we may\footnote{We omit the verification of the set theoretical condition (3) of the referenced lemma.} apply Bootstrap, Lemma \ref{bootstrap-lemma-locally-algebraic-space} to check the representability \'etale locally on $B$. Moreover, we may check whether the end result is affine or of finite presentation \'etale locally on $B$, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-affine-local} and \ref{spaces-morphisms-lemma-finite-presentation-local}. Hence we may assume that $B$ is an affine scheme. \medskip\noindent Assume $B$ is an affine scheme. As $f$ is of finite presentation, it follows $X$ is quasi-compact and quasi-separated. Thus we can write $\mathcal{F} = \colim \mathcal{F}_i$ as a filtered colimit of $\mathcal{O}_X$-modules of finite presentation (Limits of Spaces, Lemma \ref{spaces-limits-lemma-colimit-finitely-presented}). It is clear that $$ \mathit{Hom}(\mathcal{F}, \mathcal{G}) = \lim \mathit{Hom}(\mathcal{F}_i, \mathcal{G}) $$ Hence if we can show that each $\mathit{Hom}(\mathcal{F}_i, \mathcal{G})$ is representable by an affine scheme, then we see that the same thing holds for $\mathit{Hom}(\mathcal{F}, \mathcal{G})$. Use the material in Limits, Section \ref{limits-section-limits} and Limits of Spaces, Section \ref{spaces-limits-section-limits}. Thus we may assume that $\mathcal{F}$ is of finite presentation. \medskip\noindent Say $B = \Spec(R)$. Write $R = \colim R_i$ with each $R_i$ a finite type $\mathbf{Z}$-algebra. Set $B_i = \Spec(R_i)$. By the results of Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-finite-presentation} and \ref{spaces-limits-lemma-descend-modules-finite-presentation} we can find an $i$, a morphism of algebraic spaces $X_i \to B_i$, and finitely presented $\mathcal{O}_{X_i}$-modules $\mathcal{F}_i$ and $\mathcal{G}_i$ such that the base change of $(X_i, \mathcal{F}_i, \mathcal{G}_i)$ to $B$ recovers $(X, \mathcal{F}, \mathcal{G})$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-flat} we may, after increasing $i$, assume that $\mathcal{G}_i$ is flat over $B_i$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-eventually-proper-support} we may similarly assume the scheme theoretic support of $\mathcal{G}_i$ is proper over $B_i$. At this point we can apply Lemma \ref{lemma-noetherian-hom} to see that $H_i = \mathit{Hom}(\mathcal{F}_i, \mathcal{G}_i)$ is an algebraic space affine of finite presentation over $B_i$. Pulling back to $B$ (using Remark \ref{remark-hom-base-change}) we see that $H_i \times_{B_i} B = \mathit{Hom}(\mathcal{F}, \mathcal{G})$  and we win.
We will use the abbreviations $H = \mathit{Hom}(\mathcal{F}, \mathcal{G})$, $I = \mathit{Hom}(\mathcal{F}, \mathcal{F})$, $H' = \mathit{Hom}(\mathcal{G}, \mathcal{F})$, and $I' = \mathit{Hom}(\mathcal{G}, \mathcal{G})$. By Proposition \ref{proposition-hom} the functors $H$, $I$, $H'$, $I'$ are algebraic spaces and the morphisms $H \to B$, $I \to B$, $H' \to B$, and $I' \to B$ are affine and of finite presentation. The composition of maps gives a morphism $$ c : H' \times_B H \longrightarrow I \times_B I',\quad (u', u) \longmapsto (u \circ u', u' \circ u) $$ of algebraic spaces over $B$. Since $I \times_B I' \to B$ is separated, the section $\sigma : B \to I \times_B I'$ corresponding to $(\text{id}_\mathcal{F}, \text{id}_\mathcal{G})$ is a closed immersion (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion}). Moreover, $\sigma$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-permanence}). Hence $$ \mathit{Isom}(\mathcal{F}, \mathcal{G}) = (H' \times_B H) \times_{c, I \times_B I', \sigma} B $$ is an algebraic space affine of finite presentation over $B$ as well. Some details omitted.
By Lemma \ref{lemma-quot-sheaf} we have that $\Quotfunctor_{\mathcal{F}/X/B}$ is a sheaf in the fppf topology. Let $\textit{Quot}_{\mathcal{F}/X/B}$ be the stack in groupoids corresponding to $\Quotfunctor_{\mathcal{F}/X/S}$, see Algebraic Stacks, Section \ref{algebraic-section-split}. By Algebraic Stacks, Proposition \ref{algebraic-proposition-algebraic-stack-no-automorphisms} it suffices to show that $\textit{Quot}_{\mathcal{F}/X/B}$ is an algebraic stack. Consider the $1$-morphism of stacks in groupoids $$ \textit{Quot}_{\mathcal{F}/X/S} \longrightarrow \Cohstack_{X/B} $$ on $(\Sch/S)_{fppf}$ which associates to the quotient $\mathcal{F}_T \to \mathcal{Q}$ the coherent sheaf $\mathcal{Q}$. By Theorem \ref{theorem-coherent-algebraic-general} we know that $\Cohstack_{X/B}$ is an algebraic stack. By Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-morphism-to-algebraic} it suffices to show that this $1$-morphism is representable by algebraic spaces. \medskip\noindent Let $T$ be a scheme over $S$ and let the object $(h, \mathcal{G})$ of $\Cohstack_{X/B}$ over $T$ correspond to a $1$-morphism $\xi : (\Sch/T)_{fppf} \to \Cohstack_{X/B}$. The $2$-fibre product $$ \mathcal{Z} = (\Sch/T)_{fppf} \times_{\xi, \Cohstack_{X/B}} \textit{Quot}_{\mathcal{F}/X/S} $$ is a stack in setoids, see Stacks, Lemma \ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids}. The corresponding sheaf of sets (i.e., functor, see Stacks, Lemmas \ref{stacks-lemma-2-fibre-product-gives-stack-in-setoids} and \ref{stacks-lemma-when-stack-in-sets}) assigns to a scheme $T'/T$ the set of surjections $u : \mathcal{F}_{T'} \to \mathcal{G}_{T'}$ of quasi-coherent modules on $X_{T'}$. Thus we see that $\mathcal{Z}$ is representable by an open subspace (by Flatness on Spaces, Lemma \ref{spaces-flat-lemma-F-surj-open}) of the algebraic space $\mathit{Hom}(\mathcal{F}_T, \mathcal{G})$ from Proposition \ref{proposition-hom}.
There exists a surjective, flat, finitely presented morphism $B' \to B$ of algebraic spaces such that the base change $X' = X \times_B B'$ over $B'$ has a section: namely, we can take $B' = X$. Observe that $\Picardfunctor_{X'/B'} = B' \times_B \Picardfunctor_{X/B}$. Hence $\Picardfunctor_{X'/B'} \to \Picardfunctor_{X/B}$ is representable by algebraic spaces, surjective, flat, and finitely presented. Hence, if we can show that $\Picardfunctor_{X'/B'}$ is an algebraic space, then it follows that $\Picardfunctor_{X/B}$ is an algebraic space by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. In this way we reduce to the case described in the next paragraph. \medskip\noindent In addition to the assumptions of the proposition, assume that we have a section $\sigma : B \to X$. By Proposition \ref{proposition-pic} we see that $\Picardstack_{X/B}$ is an algebraic stack. By Lemma \ref{lemma-compare-pic-with-section} and Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-morphism-to-algebraic} we see that $\Picardstack_{X/B, \sigma}$ is an algebraic stack. By Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section-functor-stack} and Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-space} we see that $T \mapsto \Ker(\sigma_T^* : \Pic(X_T) \to \Pic(T))$ is an algebraic space. By Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section} this functor is the same as $\Picardfunctor_{X/B}$.
This proof repeats a few of the arguments given in the proof of the proper base change theorem for schemes. See \'Etale Cohomology, Section \ref{etale-cohomology-section-proper-base-change} for more details. \medskip\noindent The statement is \'etale local on $Y'$ and $Y$, hence we may assume both $Y$ and $Y'$ are affine schemes. Observe that this in particular proves the theorem in case $f$ is representable (we will use this below). \medskip\noindent For every $n \geq 1$ let $\mathcal{F}[n]$ be the subsheaf of sections of $\mathcal{F}$ annihilated by $n$. Then $\mathcal{F} = \colim \mathcal{F}[n]$. By  Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-colimit-cohomology} the functors $g^{-1}R^pf_*$ and $R^pf'_*(g')^{-1}$ commute with filtered colimits. Hence it suffices to prove the theorem if $\mathcal{F}$ is killed by $n$. \medskip\noindent Let $\mathcal{F} \to \mathcal{I}^\bullet$ be a resolution by injective sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Observe that $g^{-1}f_*\mathcal{I}^\bullet = f'_*(g')^{-1}\mathcal{I}^\bullet$ by Lemma \ref{lemma-proper-base-change-f-star}. Applying Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) we conclude it suffices to prove $R^pf'_*(g')^{-1}\mathcal{I}^m = 0$ for $p > 0$ and $m \in \mathbf{Z}$. \medskip\noindent Choose a surjective proper morphism $h : Z \to X$ where $Z$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. Choose an injective map $h^{-1}\mathcal{I}^m \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $Z_\etale$. Since $h$ is surjective the map $\mathcal{I}^m \to h_*\mathcal{J}$ is injective (see Lemma \ref{lemma-surjective-proper}). Since $\mathcal{I}^m$ is injective we see that $\mathcal{I}^m$ is a direct summand of $h_*\mathcal{J}$. Thus it suffices to prove the desired vanishing for $h_*\mathcal{J}$. \medskip\noindent Denote $h'$ the base change by $g$ and denote $g'' : Z' \to Z$ the projection. There is a spectral sequence $$ E_2^{p, q} = R^pf'_* R^qh'_* (g'')^{-1}\mathcal{J} $$ converging to $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J}$. Since $h$ and $f \circ h$ are representable (by schemes) we know the result we want holds for them. Thus in the spectral sequence we see that $E_2^{p, q} = 0$ for $q > 0$ and $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J} = 0$ for $p + q > 0$. It follows that $E_2^{p, 0} = 0$ for $p > 0$. Now $$ E_2^{p, 0} = R^pf'_* h'_* (g'')^{-1}\mathcal{J} = R^pf'_* (g')^{-1}h_*\mathcal{J} $$ by Lemma \ref{lemma-proper-base-change-f-star}. This finishes the proof.
For any surjective morphism $f : Y \to X$ of algebraic spaces over $S$, the map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective. Namely, if $\overline{x}$ is a geometric point of $X$, then we choose a geometric point $\overline{y}$ of $Y$ lying over $\overline{x}$ and we consider $$ \mathcal{F}_{\overline{x}} \to (f_*f^{-1}\mathcal{F})_{\overline{x}} \to (f^{-1}\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}} $$ See Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback} for the last equality. \medskip\noindent The second statement is local on $X$ in the \'etale topology, hence we may and do assume $Y$ is an affine scheme. \medskip\noindent Choose a surjective proper morphism $Z \to Y$ where $Z$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. The result for $Z \to X$ implies the result for $Y \to X$. Since $Z \to X$ is a surjective proper morphism of schemes and hence a ph covering (Topologies, Lemma \ref{topologies-lemma-surjective-proper-ph}) the result for $Z \to X$ follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-ph} (in fact it is in some sense equivalent to this lemma).
Choose a surjective proper morphism $Y \to X$ where $Y$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. Consider the diagram $$ \xymatrix{ \Gamma(X_0, \mathcal{F}_0) \ar[r] \ar[d] & \Gamma(Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[d] & \Gamma((Y \times_X Y)_0, \mathcal{H}_0) \ar[d] \\ \Gamma(X, \mathcal{F}) \ar[r] & \Gamma(Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma(Y \times_X Y, \mathcal{H}) } $$ Here $\mathcal{G}$, resp.\ $\mathcal{H}$ is the pullbackf or $\mathcal{F}$ to $Y$, resp.\ $Y \times_X Y$ and the index $0$ indicates base change to $\Spec(A/I)$. By the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-h0-proper-over-henselian-pair}) we see that the middle and right vertical arrows are bijective. By Lemma \ref{lemma-surjective-proper} it follows that the left one is too.
The question is \'etale local on $Y'$. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to V \times_Y Y'$. Then we may replace $Y'$ by $V'$ and $Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes. Then we may work Zariski locally on $Y$ and $Y'$ and hence we may assume $Y$ and $Y'$ are affine schemes. \medskip\noindent Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism $h_1 : X_1 \to X$ where $X_1$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. Set $X_2 = X_1 \times_X X_1$ and denote $h_2 : X_2 \to X$ the structure morphism. Observe this is a scheme. By the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-proper-base-change-f-star}) we know the lemma is true for the cartesian diagrams $$ \vcenter{ \xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\ Y' \ar[r] & Y } } \quad\text{and}\quad \vcenter{ \xymatrix{ X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\ Y' \ar[r] & Y } } $$ and the sheaves $\mathcal{F}_i = (X_i \to X)^{-1}\mathcal{F}$. By Lemma \ref{lemma-surjective-proper} we have an exact sequence $0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2$ and similarly for $(g')^{-1}\mathcal{F}$ because $X'_2 = X'_1 \times_{X'} X'_1$. Hence we conlude that the lemma is true (some details omitted).
This is a simple consequence of the proper base change theorem (Theorem \ref{theorem-proper-base-change}) using the spectral sequences $$ E_2^{p, q} = R^pf_*H^q(E) \quad\text{and}\quad {E'}_2^{p, q} = R^pf'_*(g')^{-1}H^q(E) $$ converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$. The spectral sequences are constructed in Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Some details omitted.
In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback of $\mathcal{F}$ to $X_{\overline{y}} = \overline{y} \times_Y X$. Since pulling back by $\overline{y} \to Y$ produces the stalk of $\mathcal{F}$, the first statement of the lemma is a special case of Theorem \ref{theorem-proper-base-change}. The second one is a special case of Lemma \ref{lemma-proper-base-change}.
Since pullback is transitive and $f_{small} = \pi_X \circ i_f$ (see above) we see that $i_f^{-1} \pi_X^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$. This shows that $\pi_X^{-1}$ has the description given in the lemma. \medskip\noindent To prove that $\pi_X^{-1}\mathcal{F}$ is a sheaf for the ph topology it suffices by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-characterize-sheaf} to show that for a surjective proper morphism $V \to U$ of algebraic spaces over $X$ we have $(\pi_X^{-1}\mathcal{F})(U)$ is the equalizer of the two maps $(\pi_X^{-1}\mathcal{F})(V) \to (\pi_X^{-1}\mathcal{F})(V \times_U V)$. This we have seen in Lemma \ref{lemma-surjective-proper}. \medskip\noindent The case of smooth, syntomic, fppf coverings follows from the case of ph coverings by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}. \medskip\noindent Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be an fpqc covering of algebraic spaces over $X$. Let $s_i \in (\pi_X^{-1}\mathcal{F})(U_i)$ be sections which agree over $U_i \times_U U_j$. We have to prove there exists a unique $s \in (\pi_X^{-1}\mathcal{F})(U)$ restricting to $s_i$ over $U_i$. Case I: $U$ and $U_i$ are schemes. This case follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback}. Case II: $U$ is a scheme. Here we choose surjective \'etale morphisms $T_i \to U_i$ where $T_i$ is a scheme. Then $\mathcal{T} = \{T_i \to U\}$ is an fpqc covering by schemes and by case I the result holds for $\mathcal{T}$. We omit the verification that this implies the result for $\mathcal{U}$. Case III: general case. Let $W \to U$ be a surjective \'etale morphism, where $W$ is a scheme. Then $\mathcal{W} = \{U_i \times_U W \to W\}$ is an fpqc covering (by algebraic spaces) of the scheme $W$. By case II the result hold for $\mathcal{W}$. We omit the verification that this implies the result for $\mathcal{U}$.