hcju/stacksqa huggingface dataset:
We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-ph-etale}. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_X$ of More on Cohomology of Spaces, Section \ref{spaces-more-cohomology-section-ph-etale}. Thus it follows from More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-comparison-ph-etale} that $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_n$ of $h$. By the description of the functors $h^{-1}$ and $h_*$ of Lemma \ref{lemma-morphism-simplicial-sites} we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$ as defined in Section \ref{section-hypercovering} since a surjective proper morphism gives a ph covering by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-surjective-proper-ph}. By Lemma \ref{lemma-hypercovering-X-simple-descent-sheaves} we see that $a_{ph}^{-1}$ is fully faithful with quasi-inverse $a_{ph, *}$ and with essential image the cartesian sheaves on $(\textit{Spaces}/U)_{ph, total}$. A formal argument (chasing around the diagram) now shows that $a^{-1}$ is fully faithful. \medskip\noindent Finally, suppose that $\mathcal{G}$ is a cartesian sheaf on $U_\etale$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $(\textit{Spaces}/U)_{ph, total}$. Hence $h^{-1}\mathcal{G} = a_{ph}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{ph}$. We compute using somewhat pedantic notation \begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}(a_{0, big, ph})^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}(a_{1, big, ph})^{-1}\mathcal{H} } ) \\ & = a_{ph, *}a_{ph}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*} Here the first equality follows from Lemma \ref{lemma-augmentation-site}, the second equality follows as $(h_{-1})^{-1}$ is an exact functor, the third equality follows from More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-proper-push-pull-ph-etale} (here we use that $a_0 : U_0 \to X$ and $a_1: U_1 \to X$ are proper), the fourth follows from $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$, the fifth from Lemma \ref{lemma-augmentation-site}, and the sixth we've seen above. Since $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$ which ends the proof by fully faithfulness of $h^{-1}$.
Consider the diagram of Lemma \ref{lemma-compare-simplicial-objects-ph-etale}. Observe that $Rh_{n, *}h_n^{-1}$ is the identity functor on $D^+(U_{n, \etale})$ by More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-cohomological-descent-etale-ph}. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_\etale)$ by Lemma \ref{lemma-direct-image-morphism-simplicial-sites}. We have \begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{ph, *}a_{ph}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*} The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma \ref{lemma-compare-simplicial-objects}, the third equality by Lemma \ref{lemma-hypercovering-X-simple-descent-bounded-abelian} as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$ by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-surjective-proper-ph}, and the last equality by the already used More on Cohomology of Spaces, Lemma \ref{spaces-more-cohomology-lemma-cohomological-descent-etale-ph}.
Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma \ref{lemma-Serre-subcat-cartesian-modules}. The equivalence is a formal consequence of the results obtained so far. Use Lemmas \ref{lemma-descent-sheaves-for-ph-hypercovering} and \ref{lemma-cohomological-descent-for-ph-hypercovering} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-equivalence-bounded}.
Let $(X, \mathcal{F}) \in \Ob(\QCohstack)$. Let $f : Y \to X$ be a morphism of $(\Sch/S)_{fppf}$. Note that there is a canonical $f$-map $c : \mathcal{F} \to f^*\mathcal{F}$ and hence we get a morphism $(f, c) : (Y, f^*\mathcal{F}) \to (X, \mathcal{F})$. We claim that $(f, c)$ is strongly cartesian. Namely, for any object $(Z, \mathcal{H})$ of $\QCohstack$ we have \begin{align*} \Mor_{\QCohstack}((Z, \mathcal{H}), (Y, f^*\mathcal{F})) & = \coprod\nolimits_{g \in \Mor_S(Z, Y)} \Mor_{\QCoh(\mathcal{O}_Z)}(g^*f^*\mathcal{F}, \mathcal{H}) \\ & = \coprod\nolimits_{g \in \Mor_S(Z, Y)} \Mor_{\QCoh(\mathcal{O}_Z)}((f \circ g)^*\mathcal{F}, \mathcal{H}) \\ & = \Mor_{\QCohstack}((Z, \mathcal{H}), (X, \mathcal{F})) \times_{\Mor_S(Z, X)} \Mor_S(Z, Y) \end{align*} where we have used Equation (\ref{equation-morphisms-qcoh}) twice. This proves that the condition of Categories, Definition \ref{categories-definition-cartesian-over-C} holds for $(f, c)$, and hence our claim is true. Now by Categories, Lemma \ref{categories-lemma-composition-cartesian} we see that isomorphisms are strongly cartesian and compositions of strongly cartesian morphisms are strongly cartesian which proves the ``if'' part of the lemma. For the converse, note that given $(X, \mathcal{F})$ and $f : Y \to X$, if there exists a strongly cartesian morphism lifting $f$ with target $(X, \mathcal{F})$ then it has to be isomorphic to $(f, c)$ (see discussion following Categories, Definition \ref{categories-definition-cartesian-over-C}). Hence the "only if" part of the lemma holds.
It is clear from Lemma \ref{lemma-quasi-coherent-strongly-cartesian} that $\QCohstack$ is a fibred category over $(\Sch/S)_{fppf}$. Given covering $\mathcal{U} = \{X_i \to X\}_{i \in I}$ of $(\Sch/S)_{fppf}$ the functor $$ \QCoh(\mathcal{O}_X) \longrightarrow DD(\mathcal{U}) $$ is fully faithful and essentially surjective, see Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. Hence Stacks, Lemma \ref{stacks-lemma-stack-equivalences} applies to show that $\QCohstack$ satisfies all the axioms of a stack.
We will verify assumptions (1), (2), (3) of Stacks, Lemma \ref{stacks-lemma-substack} to prove this. By Lemma \ref{lemma-quasi-coherent-strongly-cartesian} a morphism $(Y, \mathcal{G}) \to (X, \mathcal{F})$ is strongly cartesian if and only if it induces an isomorphism $f^*\mathcal{F} \to \mathcal{G}$. By Modules, Lemma \ref{modules-lemma-pullback-finite-type} the pullback of a finite type $\mathcal{O}_X$-module is of finite type. Hence assumption (1) of Stacks, Lemma \ref{stacks-lemma-substack} holds. Assumption (2) holds trivially. Finally, to prove assumption (3) we have to show: If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module and $\{f_i : X_i \to X\}$ is an fppf covering such that each $f_i^*\mathcal{F}$ is of finite type, then $\mathcal{F}$ is of finite type. Considering the restriction of $\mathcal{F}$ to an affine open of $X$ this reduces to the following algebra statement: Suppose that $R \to S$ is a finitely presented, faithfully flat ring map and $M$ an $R$-module. If $M \otimes_R S$ is a finitely generated $S$-module, then $M$ is a finitely generated $R$-module. A stronger form of the algebra fact can be found in Algebra, Lemma \ref{algebra-lemma-descend-properties-modules}.
We have seen in Lemmas \ref{lemma-stack-of-finite-type-quasi-coherent-sheaves} and \ref{lemma-finite-type} that $p_{fg} : \QCohstack_{fg} \to (\Sch/S)_{fppf}$ satisfies (1), (2) and (3) of Stacks, Definition \ref{stacks-definition-stack} as well as the additional condition (4) of Stacks, Remark \ref{stacks-remark-stack-make-small}. Hence we obtain $\QCohstack_{fg, small}$ from the discussion in that remark.
Let $Y/V \in \Ob(\Spacesstack)$. Let $g : U \to V$ be a morphism of $(\Sch/S)_{fppf}$. Note that the projection $p : U \times_{g, V} Y \to Y$ gives rise a morphism $(p, g) : U \times_{g, V} Y/U \to Y/V$ of $\Spacesstack$. We claim that $(p, g)$ is strongly cartesian. Namely, for any object $Z/W$ of $\Spacesstack$ we have \begin{align*} \Mor_{\Spacesstack}(Z/W, U \times_{g, V} Y/U) & = \coprod\nolimits_{h \in \Mor_S(W, U)} \Mor_{\textit{Spaces}/W}(Z, W \times_{h, U} U \times_{g, V} Y) \\ & = \coprod\nolimits_{h \in \Mor_S(W, U)} \Mor_{\textit{Spaces}/W}(Z, W \times_{g \circ h, V} Y) \\ & = \Mor_{\Spacesstack}(Z/W, Y/V) \times_{\Mor_S(W, V)} \Mor_S(W, U) \end{align*} where we have used Equation (\ref{equation-morphisms-spaces}) twice. This proves that the condition of Categories, Definition \ref{categories-definition-cartesian-over-C} holds for $(p, g)$, and hence our claim is true. Now by Categories, Lemma \ref{categories-lemma-composition-cartesian} we see that isomorphisms are strongly cartesian and compositions of strongly cartesian morphisms are strongly cartesian which proves the ``if'' part of the lemma. For the converse, note that given $Y/V$ and $g : U \to V$, if there exists a strongly cartesian morphism lifting $g$ with target $Y/V$ then it has to be isomorphic to $(p, g)$ (see discussion following Categories, Definition \ref{categories-definition-cartesian-over-C}). Hence the "only if" part of the lemma holds.
It is follows from Lemma \ref{lemma-spaces-strongly-cartesian} that $\Spacesstack$ is a fibred category over $(\Sch/S)_{fppf}$ which proves (1). Suppose that $\{U_i \to U\}_{i \in I}$ is a covering of $(\Sch/S)_{fppf}$. Suppose that $X, Y$ are algebraic spaces over $U$. Finally, suppose that $\varphi_i : X_{U_i} \to Y_{U_i}$ are morphisms of $\textit{Spaces}/U_i$ such that $\varphi_i$ and $\varphi_j$ restrict to the same morphisms $X_{U_i \times_U U_j} \to Y_{U_i \times_U U_j}$ of algebraic spaces over $U_i \times_U U_j$. To prove (2) we have to show that there exists a unique morphism $\varphi  : X \to Y$ over $U$ whose base change to $U_i$ is equal to $\varphi_i$. As a morphism from $X$ to $Y$ is the same thing as a map of sheaves this follows directly from Sites, Lemma \ref{sites-lemma-glue-maps}.
We are going to write this out in ridiculous detail (which may make it hard to see what is going on). \medskip\noindent We have seen in Lemma \ref{lemma-spaces-strongly-cartesian} that a morphism $(f, g) : X/U \to Y/V$ of $\Spacesstack$ is strongly cartesian if the induced morphism $f : X \to U \times_V Y$ is an isomorphism. Note that if $Y \to V$ is of finite type then also $U \times_V Y \to U$ is of finite type, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-type}. So if $(f, g) : X/U \to Y/V$ of $\Spacesstack$ is strongly cartesian in $\Spacesstack$ and $Y/V$ is an object of $\Spacesstack_{ft}$ then automatically also $X/U$ is an object of $\Spacesstack_{ft}$, and of course $(f, g)$ is also strongly cartesian in $\Spacesstack_{ft}$. In this way we conclude that $\Spacesstack_{ft}$ is a fibred category over $(\Sch/S)_{fppf}$. This proves (1). \medskip\noindent The argument above also shows that the inclusion functor $\Spacesstack_{ft} \to \Spacesstack$ transforms strongly cartesian morphisms into strongly cartesian morphisms. In other words $\Spacesstack_{ft} \to \Spacesstack$ is a $1$-morphism of fibred categories over $(\Sch/S)_{fppf}$. \medskip\noindent Let $U \in \Ob((\Sch/S)_{fppf})$. Let $X, Y$ be algebraic spaces of finite type over $U$. By Stacks, Lemma \ref{stacks-lemma-presheaf-mor-map-fibred-categories} we obtain a map of presheaves $$ \mathit{Mor}_{\Spacesstack_{ft}}(X, Y) \longrightarrow \mathit{Mor}_{\Spacesstack}(X, Y) $$ which is an isomorphism as $\Spacesstack_{ft}$ is a full subcategory of $\Spacesstack$. Hence the left hand side is a sheaf, because in Lemma \ref{lemma-pre-stack-of-spaces} we showed the right hand side is a sheaf. This proves (2). \medskip\noindent To prove condition (3) of Stacks, Definition \ref{stacks-definition-stack} we have to show the following: Given \begin{enumerate} \item a covering $\{U_i \to U\}_{i \in I}$ of $(\Sch/S)_{fppf}$, \item for each $i \in I$ an algebraic space $X_i$ of finite type over $U_i$, and \item for each $i, j \in I$ an isomorphism $\varphi_{ij} : X_i \times_U U_j \to U_i \times_U X_j$ of algebraic spaces over $U_i \times_U U_j$ satisfying the cocycle condition over $U_i \times_U U_j \times_U U_k$, \end{enumerate} there exists an algebraic space $X$ of finite type over $U$ and isomorphisms $X_{U_i} \cong X_i$ over $U_i$ recovering the isomorphisms $\varphi_{ij}$. This follows from Bootstrap, Lemma \ref{bootstrap-lemma-descend-algebraic-space} part (2). By Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-locally-finite-presentation} we see that $X \to U$ is of finite type which concludes the proof.
We have seen in Lemmas \ref{lemma-stack-of-finite-type-spaces} that $p_{ft} : \Spacesstack_{ft} \to (\Sch/S)_{fppf}$ satisfies (1), (2) and (3) of Stacks, Definition \ref{stacks-definition-stack}. The additional condition (4) of Stacks, Remark \ref{stacks-remark-stack-make-small} holds because every algebraic space $X$ over $S$ is of the form $U/R$ for $U, R \in \Ob((\Sch/S)_{fppf})$, see Spaces, Lemma \ref{spaces-lemma-space-presentation}. Thus there is only a set worth of isomorphism classes of objects. Hence we obtain $\Spacesstack_{ft, small}$ from the discussion in that remark.
The most difficult part of the proof is to show that we have descent for objects. Let $\{U_i \to U\}_{i \in I}$ be a covering of $(\Sch/S)_{fppf}$. Suppose that for each $i$ we are given a $\mathcal{G}|_{U_i}$-torsor $\mathcal{F}_i$, and for each $i, j \in I$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \times_U U_j} \to \mathcal{F}_j|_{U_i \times_U U_j}$ of $\mathcal{G}|_{U_i \times_U U_j}$-torsors satisfying a suitable cocycle condition on $U_i \times_U U_j \times_U U_k$. Then by Sites, Section \ref{sites-section-glueing-sheaves} we obtain a sheaf $\mathcal{F}$ on $(\Sch/U)_{fppf}$ whose restriction to each $U_i$ recovers $\mathcal{F}_i$ as well as recovering the descent data. By the equivalence of categories in Sites, Lemma \ref{sites-lemma-mapping-property-glue} the action maps $\mathcal{G}|_{U_i} \times \mathcal{F}_i \to \mathcal{F}_i$ glue to give a map $a : \mathcal{G}|_U \times \mathcal{F} \to \mathcal{F}$. Now we have to show that $a$ is an action and that $\mathcal{F}$ becomes a $\mathcal{G}|_U$-torsor. Both properties may be checked locally, and hence follow from the corresponding properties of the actions $\mathcal{G}|_{U_i} \times \mathcal{F}_i \to \mathcal{F}_i$. This proves that descent for objects holds in $\mathcal{G}\textit{-Torsors}$. Some details omitted.
This proof is a repeat of the proof of Lemma \ref{lemma-torsors-sheaf-stack-in-groupoids}. The reader is encouraged to read that proof first since the notation is less cumbersome. The most difficult part of the proof is to show that we have descent for objects. Let $\{U_i \to U\}_{i \in I}$ be a covering of $(\Sch/S)_{fppf}$. Suppose that for each $i$ we are given a pair $(b_i, \mathcal{F}_i)$ consisting of a morphism $b_i : U_i \to \mathcal{B}$ and a $U_i \times_{b_i, \mathcal{B}} \mathcal{G}$-torsor $\mathcal{F}_i$, and for each $i, j \in I$ we have $b_i|_{U_i \times_U U_j} = b_j|_{U_i \times_U U_j}$ and we are given an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \times_U U_j} \to \mathcal{F}_j|_{U_i \times_U U_j}$ of $(U_i \times_U U_j) \times_\mathcal{B} \mathcal{G}$-torsors satisfying a suitable cocycle condition on $U_i \times_U U_j \times_U U_k$. Then by Sites, Section \ref{sites-section-glueing-sheaves} we obtain a sheaf $\mathcal{F}$ on $(\Sch/U)_{fppf}$ whose restriction to each $U_i$ recovers $\mathcal{F}_i$ as well as recovering the descent data. By the sheaf axiom for $\mathcal{B}$ the morphisms $b_i$ come from a unique morphism $b : U \to \mathcal{B}$. By the equivalence of categories in Sites, Lemma \ref{sites-lemma-mapping-property-glue} the action maps $(U_i \times_{b_i, \mathcal{B}} \mathcal{G}) \times_{U_i} \mathcal{F}_i \to \mathcal{F}_i$ glue to give a map $(U \times_{b, \mathcal{B}} \mathcal{G}) \times \mathcal{F} \to \mathcal{F}$. Now we have to show that this is an action and that $\mathcal{F}$ becomes a $U \times_{b, \mathcal{B}} \mathcal{G}$-torsor. Both properties may be checked locally, and hence follow from the corresponding properties of the actions on the $\mathcal{F}_i$. This proves that descent for objects holds in $\mathcal{G}/\mathcal{B}\textit{-Torsors}$. Some details omitted.
We will use the result of Stacks, Lemma \ref{stacks-lemma-characterize-essentially-surjective-when-ff} to prove this. The functor is fully faithful since the category of algebraic spaces over $S$ is a full subcategory of the category of sheaves on $(\Sch/S)_{fppf}$. Moreover, all objects (on both sides) are locally trivial torsors so condition (2) of the lemma referenced above holds. Hence the functor is an equivalence.
The most difficult part of the proof is to show that we have descent for objects. Suppose that $\{U_i \to U\}_{i \in I}$ is a covering in $(\Sch/S)_{fppf}$. Let $\xi_i = (U_i, b_i, P_i, \varphi_i)$ be objects of $[[X/G]]$ over $U_i$, and let $\varphi_{ij} : \text{pr}_0^*\xi_i \to \text{pr}_1^*\xi_j$ be a descent datum. This in particular implies that we get a descent datum on the triples $(U_i, b_i, P_i)$ for the stack in groupoids $G\textit{-Torsors}$ by applying the functor $[[X/G]] \to G\textit{-Torsors}$ above. We have seen that $G\textit{-Torsors}$ is a stack in groupoids (Lemma \ref{lemma-torsors-stack-in-groupoids}). Hence we may assume that $b_i = b|_{U_i}$ for some morphism $b : U \to B$, and that $P_i = U_i \times_U P$ for some fppf $G_U = U \times_{b, B} G$-torsor $P$ over $U$. The morphisms $\varphi_i$ are compatible with the canonical descent datum on the restrictions $U_i \times_U P$ and hence define a morphism $\varphi : P \to X$. (For example you can use Sites, Lemma \ref{sites-lemma-mapping-property-glue} or you can use Descent on Spaces, Lemma \ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms} to get $\varphi$.) This proves descent for objects. We omit the proof of axioms (1) and (2) of Stacks, Definition \ref{stacks-definition-stack-in-groupoids}.
The equivalence $G\textit{-Torsors} \to \mathcal{G}/\mathcal{B}\textit{-Torsors}$ is given in Lemma \ref{lemma-compare-torsors}. The equivalence $[B/G] \to [[B/G]]$ is given in Proposition \ref{proposition-equal-quotient-stacks}. Unwinding the definition of $[[B/G]]$ given in Section \ref{section-group-quotient-stacks} we see that $[[B//G]] = G\textit{-Torsors}$. \medskip\noindent Finally, assume $G \to B$ is flat and locally of finite presentation. To show that the natural functor $G\textit{-Torsors} \to G\textit{-Principal}$ is an equivalence it suffices to show that for a scheme $U$ over $B$ a principal homogeneous $G_U$-space $X \to U$ is fppf locally trivial. By our definition of principal homogeneous spaces (Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-principal-homogeneous-space}) there exists an fpqc covering $\{U_i \to U\}$ such that $U_i \times_U X \cong G \times_B U_i$ as algebraic spaces over $U_i$. This implies that $X \to U$ is surjective, flat, and locally of finite presentation, see Descent on Spaces, Lemmas \ref{spaces-descent-lemma-descending-property-surjective}, \ref{spaces-descent-lemma-descending-property-flat}, and \ref{spaces-descent-lemma-descending-property-locally-finite-presentation}. Choose a scheme $W$ and a surjective \'etale morphism $W \to X$. Then it follows from what we just said that $\{W \to U\}$ is an fppf covering such that $X_W \to W$ has a section. Hence $X$ is an fppf $G_U$-torsor.
As usual, the hardest part is to show descent for objects. To see this let $\{U_i \to U\}$ be a covering of $(\Sch/S)_{fppf}$. Let $\xi_i = (U_i, b_i, \mathcal{L}_i)$ be an object of $\Picardstack_{X/B}$ lying over $U_i$, and let $\varphi_{ij} : \text{pr}_0^*\xi_i \to \text{pr}_1^*\xi_j$ be a descent datum. This implies in particular that the morphisms $b_i$ are the restrictions of a morphism $b : U \to B$. Write $X_U = U \times_{b, B} X$ and $X_i = U_i \times_{b_i, B} X = U_i \times_U U \times_{b, B} X = U_i \times_U X_U$. Observe that $\mathcal{L}_i$ is an invertible $\mathcal{O}_{X_i}$-module. Note that $\{X_i \to X_U\}$ forms an fppf covering as well. Moreover, the descent datum $\varphi_{ij}$ translates into a descent datum on the invertible sheaves $\mathcal{L}_i$ relative to the fppf covering $\{X_i \to X_U\}$. Hence by Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we obtain a unique invertible sheaf $\mathcal{L}$ on $X_U$ which recovers $\mathcal{L}_i$ and the descent data over $X_i$. The triple $(U, b, \mathcal{L})$ is therefore the object of $\Picardstack_{X/B}$ over $U$ we were looking for. Details omitted.
As usual, the hardest part is to show descent for objects. To see this let $\{U_i \to U\}$ be a covering of $(\Sch/S)_{fppf}$. Let $\xi_i = (U_i, Z_i, y_i, x_i, \alpha_i)$ be an object of $\mathcal{H}_d(\mathcal{X}/\mathcal{Y})$ lying over $U_i$, and let $\varphi_{ij} : \text{pr}_0^*\xi_i \to \text{pr}_1^*\xi_j$ be a descent datum. First, observe that $\varphi_{ij}$ induces a descent datum $(Z_i/U_i, \varphi_{ij})$ which is effective by Descent, Lemma \ref{descent-lemma-affine} This produces a scheme $Z/U$ which is finite locally free of degree $d$ by Descent, Lemma \ref{descent-lemma-descending-property-finite-locally-free}. From now on we identify $Z_i$ with $Z \times_U U_i$. Next, the objects $y_i$ in the fibre categories $\mathcal{Y}_{U_i}$ descend to an object $y$ in $\mathcal{Y}_U$ because $\mathcal{Y}$ is a stack in groupoids. Similarly the objects $x_i$ in the fibre categories $\mathcal{X}_{Z_i}$ descend to an object $x$ in $\mathcal{X}_Z$ because $\mathcal{X}$ is a stack in groupoids. Finally, the given isomorphisms $$ \alpha_i : (y|_Z)_{Z_i} = y_i|_{Z_i} \longrightarrow F(x_i) = F(x|_{Z_i}) $$ glue to a morphism $\alpha : y|_Z \to F(x)$ as the $\mathcal{Y}$ is a stack and hence $\mathit{Isom}_\mathcal{Y}(y|_Z, F(x))$ is a sheaf. Details omitted.
For the case of genus zero, see Section \ref{section-semistable-reduction-genus-zero}. For the case of genus one, see Section \ref{section-semistable-reduction-genus-one}. For the case of genus greater than one, see Section \ref{section-semistable-reduction-genus-at-least-two}. To see that we have a bound on the degree $[K' : K]$ you can use the bound on the degree of the extension needed to make all $\ell$ or $\ell'$ torsion visible proved in Algebraic Curves, Lemma \ref{curves-lemma-torsion-picard-becomes-visible}. (The reason for using $\ell$ and $\ell'$ is that we need to avoid the characteristic of the residue field $k$.)