hcju/stacksqa huggingface dataset:
In Lemma \ref{lemma-q-sheaf} we have seen that the functor $\text{Q}^{fp}_{\mathcal{F}/X/S}$ is a sheaf. Recall that for a scheme $T$ over $S$ the subset $\Quotfunctor_{\mathcal{F}/X/S}(T) \subset \text{Q}_{\mathcal{F}/X/S}(T)$ picks out those quotients whose support is proper over $T$. This defines a subsheaf by the result of Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-proper} combined with Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-pullback-support} which shows that taking scheme theoretic support commutes with flat base change.
Observe that the left and right hand side of the equality are subsets of the left and right hand side of the second equality in Lemma \ref{lemma-extend-q-to-spaces}. To see that these subsets correspond under the identification given in the proof of that lemma it suffices to show: given $h : T \to B$, a surjective \'etale morphism $U \to T$, a finite type quasi-coherent $\mathcal{O}_{X_T}$-module $\mathcal{Q}$ the following are equivalent \begin{enumerate} \item the scheme theoretic support of $\mathcal{Q}$ is proper over $T$, and \item the scheme theoretic support of $(X_U \to X_T)^*\mathcal{Q}$ is proper over $U$. \end{enumerate} This follows from Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-proper} combined with Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-pullback-support} which shows that taking scheme theoretic support commutes with flat base change.
Let $T$ be a scheme over $B$. Given an element $Z \in \Hilbfunctor_{X/B}(T)$ we can consider the quotient $\mathcal{O}_{X_T} \to i_*\mathcal{O}_Z$ where $i : Z \to X_T$ is the inclusion morphism. Note that $i_*\mathcal{O}_Z$ is quasi-coherent. Since $Z \to T$ and $X_T \to T$ are of finite presentation, we see that $i$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-permanence}), hence $i_*\mathcal{O}_Z$ is an $\mathcal{O}_{X_T}$-module of finite presentation (Descent on Spaces, Lemma \ref{spaces-descent-lemma-finite-finitely-presented-module}). Since $Z \to T$ is proper we see that $i_*\mathcal{O}_Z$ has support proper over $T$ (as defined in Derived Categories of Spaces, Section \ref{spaces-perfect-section-proper-over-base}). Since $\mathcal{O}_Z$ is flat over $T$ and $i$ is affine, we see that $i_*\mathcal{O}_Z$ is flat over $T$ (small argument omitted). Hence $\mathcal{O}_{X_T} \to i_*\mathcal{O}_Z$ is an element of $\Quotfunctor_{\mathcal{O}_X/X/B}(T)$. \medskip\noindent Conversely, given an element $\mathcal{O}_{X_T} \to \mathcal{Q}$ of $\Quotfunctor_{\mathcal{O}_X/X/B}(T)$, we can consider the closed immersion $i : Z \to X_T$ corresponding to the quasi-coherent ideal sheaf $\mathcal{I} = \Ker(\mathcal{O}_{X_T} \to \mathcal{Q})$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}). By construction of $Z$ we see that $\mathcal{Q} = i_*\mathcal{O}_Z$. Then we can read the arguments given above backwards to see that $Z$ defines an element of $\Hilbfunctor_{X/B}(T)$. For example, $\mathcal{I}$ is quasi-coherent of finite type (Modules on Sites, Lemma \ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}) hence $i : Z \to X_T$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}) hence $Z \to T$ is of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-presentation}). Properness of $Z \to T$ follows from the discussion in Derived Categories of Spaces, Section \ref{spaces-perfect-section-proper-over-base}. Flatness of $Z \to T$ follows from flatness of $\mathcal{Q}$ over $T$. \medskip\noindent We omit the (immediate) verification that the two constructions given above are mutually inverse.
By Lemma \ref{lemma-hilb-is-quot} we have $\Hilbfunctor_{X/B} = \Quotfunctor_{\mathcal{O}_X/X/B}$. Thus we can apply Lemma \ref{lemma-extend-quot-to-spaces} to see that the left hand side is bijective with the set of surjections $\mathcal{O}_{X_T} \to \mathcal{Q}$ which are finitely presented, flat over $T$, and have support proper over $T$. Arguing exactly as in the proof of Lemma \ref{lemma-hilb-is-quot} we see that such quotients correspond exactly to the closed immersions $Z \to X_T$ such that $Z \to T$ is proper, flat, and of finite presentation.
Observe that the map simply sends a triple $(T, g, \mathcal{L})$ as in Examples of Stacks, Section \ref{examples-stacks-section-picard-stack} to the same triple $(T, g, \mathcal{L})$ but where now we view this as a triple of the kind described in Situation \ref{situation-coherent}. This works because the invertible $\mathcal{O}_{X_T}$-module $\mathcal{L}$ is certainly a finitely presented $\mathcal{O}_{X_T}$-module, it is flat over $T$ because $X_T \to T$ is flat, and the support is proper over $T$ as $X_T \to T$ is proper (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-flat} and \ref{spaces-morphisms-lemma-base-change-proper}). Thus the statement makes sense. \medskip\noindent Having said this, it is clear that the content of the lemma is the following: given an object $(T, g, \mathcal{F})$ of $\Cohstack_{X/B}$ there is an open subscheme $U \subset T$ such that for a morphism of schemes $T' \to T$ the following are equivalent \begin{enumerate} \item[(a)] $T' \to T$ factors through $U$, \item[(b)] the pullback $\mathcal{F}_{T'}$ of $\mathcal{F}$ by $X_{T'} \to X_T$ is invertible. \end{enumerate} Let $W \subset |X_T|$ be the set of points $x \in |X_T|$ such that $\mathcal{F}$ is locally free in a neighbourhood of $x$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-finite-free-open}. $W$ is open and formation of $W$ commutes with arbitrary base change. Clearly, if $T' \to T$ satisfies (b), then $|X_{T'}| \to |X_T|$ maps into $W$. Hence we may replace $T$ by the open $T \setminus f_T(|X_T| \setminus W)$ in order to construct $U$. After doing so we reach the situation where $\mathcal{F}$ is finite locally free. In this case we get a disjoint union decomposition $X_T = X_0 \amalg X_1 \amalg X_2 \amalg \ldots$ into open and closed subspaces such that the restriction of $\mathcal{F}$ is locally free of rank $i$ on $X_i$. Then clearly $$ U = T \setminus f_T(|X_0| \cup |X_2| \cup |X_3| \cup \ldots ) $$ works. (Note that if we assume that $T$ is quasi-compact, then $X_T$ is quasi-compact hence only a finite number of $X_i$ are nonempty and so $U$ is indeed open.)
We may replace $B$ by $T$ and $X$ by $X_T$ and assume that $B = T$ to simplify the notation. Let $\mathcal{N}$ be an invertible $\mathcal{O}_B$-module. If $f^*\mathcal{N} \cong \mathcal{O}_X$, then we see that $f_*f^*\mathcal{N} \cong f_*\mathcal{O}_X \cong \mathcal{O}_B$ by assumption. Since $\mathcal{N}$ is locally trivial, we see that the canonical map $\mathcal{N} \to f_*f^*\mathcal{N}$ is locally an isomorphism (because $\mathcal{O}_B \to f_*f^*\mathcal{O}_B$ is an isomorphism by assumption). Hence we conclude that $\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_B$ is an isomorphism and we see that $\mathcal{N}$ is trivial. This proves the first arrow is injective. \medskip\noindent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module which is in the kernel of $\Pic(X) \to \Picardfunctor_{X/B}(B)$. Then there exists an fppf covering $\{B_i \to B\}$ such that $\mathcal{L}$ pulls back to the trivial invertible sheaf on $X_{B_i}$. Choose a trivializing section $s_i$. Then $\text{pr}_0^*s_i$ and $\text{pr}_1^*s_j$ are both trivialising sections of $\mathcal{L}$ over $X_{B_i \times_B B_j}$ and hence differ by a multiplicative unit $$ f_{ij} \in \Gamma(X_{S_i \times_B B_j}, \mathcal{O}_{X_{B_i \times_B B_j}}^*) = \Gamma(B_i \times_B B_j, \mathcal{O}_{B_i \times_N B_j}^*) $$ (equality by our assumption on pushforward of structure sheaves). Of course these elements satisfy the cocycle condition on $B_i \times_B B_j \times_B B_k$, hence they define a descent datum on invertible sheaves for the fppf covering $\{B_i \to B\}$. By Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_B$-module $\mathcal{N}$ with trivializations over $B_i$ whose associated descent datum is $\{f_{ij}\}$. (The proposition applies because $B$ is a scheme by the replacement performed at the start of the proof.) Then $f^*\mathcal{N} \cong \mathcal{L}$ as the functor from descent data to modules is fully faithful.
Denote $K(T) = \Ker(\sigma_T^* : \Pic(X_T) \to \Pic(T))$. Since $\sigma$ is a section of $f$ we see that $\Pic(X_T)$ is the direct sum of $\Pic(T)$ and $K(T)$. Thus by Lemma \ref{lemma-flat-geometrically-connected-fibres} we see that $K(T) \subset \Picardfunctor_{X/B}(T)$ for all $T$. Moreover, it is clear from the construction that $\Picardfunctor_{X/B}$ is the sheafification of the presheaf $K$. To finish the proof it suffices to show that $K$ satisfies the sheaf condition for fppf coverings which we do in the next paragraph. \medskip\noindent Let $\{T_i \to T\}$ be an fppf covering. Let $\mathcal{L}_i$ be elements of $K(T_i)$ which map to the same elements of $K(T_i \times_T T_j)$ for all $i$ and $j$. Choose an isomorphism $\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i$ for all $i$. Choose an isomorphism $$ \varphi_{ij} : \mathcal{L}_i|_{X_{T_i \times_T T_j}} \longrightarrow \mathcal{L}_j|_{X_{T_i \times_T T_j}} $$ If the map $$ \alpha_j|_{T_i \times_T T_j} \circ \sigma_{T_i \times_T T_j}^*\varphi_{ij} \circ \alpha_i|_{T_i \times_T T_j} : \mathcal{O}_{T_i \times_T T_j} \to \mathcal{O}_{T_i \times_T T_j} $$ is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale $\varphi_{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider the self map $$ \varphi_{ki}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{jk}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{ij}|_{X_{T_i \times_T T_j \times_T T_k}} \quad\text{on}\quad \mathcal{L}_i|_{X_{T_i \times_T T_j \times_T T_k}} $$ which is given by multiplication by some section $f_{ijk}$ of the structure sheaf of $X_{T_i \times_T T_j \times_T T_k}$. By our choice of $\varphi_{ij}$ we see that the pullback of this map by $\sigma$ is equal to multiplication by $1$. By our assumption on functions on $X$, we see that $f_{ijk} = 1$. Thus we obtain a descent datum for the fppf covering $\{X_{T_i} \to X\}$. By Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_{X_T}$-module $\mathcal{L}$ and an isomorphism $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ whose pullback to $X_{T_i}$ recovers $(\mathcal{L}_i, \alpha_i)$ (small detail omitted). Thus $\mathcal{L}$ defines an object of $K(T)$ as desired.
We already know that $\Picardstack_{X/B}$ is a stack in groupoids over $(\Sch/S)_{fppf}$ by Examples of Stacks, Lemma \ref{examples-stacks-lemma-picard-stack}. Let us show descent for objects for $\Picardstack_{X/B, \sigma}$. Let $\{T_i \to T\}$ be an fppf covering and let $\xi_i = (T_i, h_i, \mathcal{L}_i, \alpha_i)$ be an object of $\Picardstack_{X/B, \sigma}$ lying over $T_i$, and let $\varphi_{ij} : \text{pr}_0^*\xi_i \to \text{pr}_1^*\xi_j$ be a descent datum. Applying the result for $\Picardstack_{X/B}$ we see that we may assume we have an object $(T, h, \mathcal{L})$ of $\Picardstack_{X/B}$ over $T$ which pulls back to $\xi_i$ for all $i$. Then we get $$ \alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i = (T_i \to T)^*\sigma_T^*\mathcal{L} $$ Since the maps $\varphi_{ij}$ are compatible with the $\alpha_i$ we see that $\alpha_i$ and $\alpha_j$ pullback to the same map on $T_i \times_T T_j$. By descent of quasi-coherent sheaves (Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}, we see that the $\alpha_i$ are the restriction of a single map $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ as desired. We omit the proof of descent for morphisms.
Let $T$ be a scheme and let $(\Sch/T)_{fppf} \to \Picardstack_{X/B}$ be given by the object $\xi = (T, h, \mathcal{L})$ of $\Picardstack_{X/B}$ over $T$. We have to show that $$ (\Sch/T)_{fppf} \times_{\xi, \Picardstack_{X/B}} \Picardstack_{X/B, \sigma} $$ is representable by a scheme $V$ and that the corresponding morphism $V \to T$ is surjective and smooth. See Algebraic Stacks, Sections \ref{algebraic-section-representable-morphism}, \ref{algebraic-section-morphisms-representable-by-algebraic-spaces}, and \ref{algebraic-section-representable-properties}. The forgetful functor $\Picardstack_{X/B, \sigma} \to \Picardstack_{X/B}$ is faithful on fibre categories and for $T'/T$ the set of isomorphism classes is the set of isomorphisms $$ \alpha' : \mathcal{O}_{T'} \longrightarrow (T' \to T)^*\sigma_T^*\mathcal{L} $$ See Algebraic Stacks, Lemma \ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids}. We know this functor is representable by an affine scheme $U$ of finite presentation over $T$ by Proposition \ref{proposition-isom} (applied to $\text{id} : T \to T$ and $\mathcal{O}_T$ and $\sigma^*\mathcal{L}$). Working Zariski locally on $T$ we may assume that $\sigma_T^*\mathcal{L}$ is isomorphic to $\mathcal{O}_T$ and then we see that our functor is representable by $\mathbf{G}_m \times T$ over $T$. Hence $U \to T$ Zariski locally on $T$ looks like  the projection $\mathbf{G}_m \times T \to T$ which is indeed smooth and surjective.
Let $T$ be a scheme over $B$ and let $s, t \in \Picardfunctor_{X/B}(T)$. We want to show that there exists a locally closed subscheme $Z \subset T$ such that $s|_Z = t|_Z$ and such that a morphism $T' \to T$ factors through $Z$ if and only if $s|_{T'} = t|_{T'}$. \medskip\noindent We first reduce the general problem to the case where $s$ and $t$ come from invertible modules on $X_T$. We suggest the reader skip this step. Choose an fppf covering $\{T_i \to T\}_{i \in I}$ such that $s|_{T_i}$ and $t|_{T_i}$ come from $\Pic(X_{T_i})$ for all $i$. Suppose that we can show the result for all the pairs $s|_{T_i}, t|_{T_i}$. Then we obtain locally closed subschemes $Z_i \subset T_i$ with the desired universal property. It follows that $Z_i$ and $Z_j$ have the same scheme theoretic inverse image in $T_i \times_T T_j$. This determines a descend datum on $Z_i/T_i$. Since $Z_i \to T_i$ is locally quasi-finite, it follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} that we obtain a locally quasi-finite morphism $Z \to T$ recovering $Z_i \to T_i$ by base change. Then $Z \to T$ is an immersion by Descent, Lemma \ref{descent-lemma-descending-fppf-property-immersion}. Finally, because $\Picardfunctor_{X/B}$ is an fppf sheaf, we conclude that $s|_Z = t|_Z$ and that $Z$ satisfies the universal property mentioned above. \medskip\noindent Assume $s$ and $t$ come from invertible modules $\mathcal{V}$, $\mathcal{W}$ on $X_T$. Set $\mathcal{L} = \mathcal{V} \otimes \mathcal{W}^{\otimes -1}$ We are looking for a locally closed subscheme $Z$ of $T$ such that $T' \to T$ factors through $Z$ if and only if $\mathcal{L}_{X_{T'}}$ is the pullback of an invertible sheaf on $T'$, see Lemma \ref{lemma-flat-geometrically-connected-fibres}. Hence the existence of $Z$ follows from More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-diagonal-picard-flat-proper}.
Given a scheme $T$ over $B$ and a morphism $f_T : Z_T \to X_T$ over $T$, the graph of $f$ is the morphism $\Gamma_f = (\text{id}, f) : Z_T \to Z_T \times_T X_T = (Z \times_B X)_T$. Recall that being separated, flat, proper, or finite presentation are properties of morphisms of algebraic spaces which are stable under base change (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-separated}, \ref{spaces-morphisms-lemma-base-change-flat}, \ref{spaces-morphisms-lemma-base-change-proper}, and \ref{spaces-morphisms-lemma-base-change-finite-presentation}). Hence $\Gamma_f$ is a closed immersion by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-semi-diagonal}. Moreover, $\Gamma_f(Z_T)$ is flat, proper, and of finite presentation over $T$. Thus $\Gamma_f(Z_T)$ defines an element of $\Hilbfunctor_{Z \times_B X/B}(T)$. To show the transformation is injective it suffices to show that two morphisms with the same graph are the same. This is true because if $Y \subset (Z \times_B X)_T$ is the graph of a morphism $f$, then we can recover $f$ by using the inverse of $\text{pr}_1|_Y : Y \to Z_T$ composed with $\text{pr}_2|_Y$.
Let $T$ be a scheme over $B$ and let $Y \subset (Z \times_B X)_T$ be an element of $\Hilbfunctor_{Z \times_B X/B}(T)$. Then we see that $Y$ is the graph of a morphism $Z_T \to X_T$ over $T$ if and only if $k = \text{pr}_1|_Y : Y \to Z_T$ is an isomorphism. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-where-isomorphism} there exists an open subscheme $V \subset T$ such that for any morphism of schemes $T' \to T$ we have $k_{T'} : Y_{T'} \to Z_{T'}$ is an isomorphism if and only if $T' \to T$ factors through $V$. This proves the lemma.
We have seen this in Examples of Stacks, Section \ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces} for the case of $\Spacesstack'_{ft}$ and this easily implies the result for the other case. However, let us also prove this directly by checking conditions (1) and (2) of Categories, Definition \ref{categories-definition-fibred-groupoids}. \medskip\noindent Condition (1). Let $X \to S$ be an object of $\Spacesstack'_{ft}$ and let $S' \to S$ be a morphism of schemes. Then we set $X' = S' \times_S X$. Note that $X' \to S'$ is of finite type by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-type}. to obtain a morphism $(X' \to S') \to (X \to S)$ lying over $S' \to S$. Argue similarly for the other case using Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-finite-presentation}, \ref{spaces-morphisms-lemma-base-change-flat}, and \ref{spaces-morphisms-lemma-base-change-proper}. \medskip\noindent Condition (2). Consider morphisms $(f, g) : (X' \to S') \to (X \to S)$ and $(a, b) : (Y \to T) \to (X \to S)$ of $\Spacesstack'_{ft}$. Given a morphism $h : T \to S'$ with $g \circ h = b$ we have to show there is a unique morphism $(k, h) : (Y \to T) \to (X' \to S')$ of $\Spacesstack'_{ft}$ such that $(f, g) \circ (k, h) = (a, b)$. This is clear from the fact that $X' = S' \times_S X$. The same therefore works for any full subcategory of $\Spacesstack'_{ft}$ satisfying (1).
We will use criterion (2) of Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. Let $S$ be a scheme and let $X$ and $Y$ be algebraic spaces of finite presentation over $S$, flat over $S$, and proper over $S$. We have to show that the functor $$ \mathit{Isom}_S(X, Y) : (\Sch/S)_{fppf} \longrightarrow \textit{Sets}, \quad T \longmapsto \{f : X_T \to Y_T \text{ isomorphism}\} $$ is an algebraic space. An elementary argument shows that $\mathit{Isom}_S(X, Y)$ sits in a fibre product $$ \xymatrix{ \mathit{Isom}_S(X, Y) \ar[r] \ar[d] & S \ar[d]_{(\text{id}, \text{id})} \\ \mathit{Mor}_S(X, Y) \times \mathit{Mor}_S(Y, X) \ar[r] & \mathit{Mor}_S(X, X) \times \mathit{Mor}_S(Y, Y) } $$ The bottom arrow sends $(\varphi, \psi)$ to $(\psi \circ \varphi, \varphi \circ \psi)$. By Proposition \ref{proposition-Mor} the functors on the bottom row are algebraic spaces over $S$.  Hence the result follows from the fact that the category of algebraic spaces over $S$ has fibre products.
The reason this lemma holds is the slogan: any fppf descent datum for algebraic spaces is effective, see Bootstrap, Section \ref{bootstrap-section-applications}. More precisely, the lemma for $\Spacesstack'_{ft}$ follows from Examples of Stacks, Lemma \ref{examples-stacks-lemma-stack-of-finite-type-spaces} as we saw in Examples of Stacks, Section \ref{examples-stacks-section-stack-in-groupoids-of-finite-type-spaces}. However, let us review the proof. We need to check conditions (1), (2), and (3) of Stacks, Definition \ref{stacks-definition-stack-in-groupoids}. \medskip\noindent Property (1) we have seen in Lemma \ref{lemma-spaces-fibred-in-groupoids}. \medskip\noindent Property (2) follows from Lemma \ref{lemma-spaces-diagonal} in the case of $\Spacesstack'_{fp, flat, proper}$. In the case of $\Spacesstack'_{ft}$ it follows from Examples of Stacks, Lemma \ref{examples-stacks-lemma-pre-stack-of-spaces} (and this is really the ``correct'' reference). \medskip\noindent Condition (3) for $\Spacesstack'_{ft}$ is checked as follows. Suppose given \begin{enumerate} \item an fppf covering $\{U_i \to U\}_{i \in I}$ in $\Sch_{fppf}$, \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} We have to show 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-finite-type} we see that $X \to U$ is of finite type. In the case of $\Spacesstack'_{fp, flat, proper}$ one additionally uses Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-finite-presentation}, \ref{spaces-descent-lemma-descending-property-flat}, and \ref{spaces-descent-lemma-descending-property-proper} in the last step.
We are going to deduce this lemma from the fact that it holds for schemes (essentially by construction of the stacks) and the fact that fppf descent data for algebraic spaces over algerbaic spaces are effective. We strongly encourage the reader to skip the proof. \medskip\noindent The construction from left to right in either arrow is straightforward: given $X \to T$ of finite type the functor $\mathcal{S}_T \to  \Spacesstack'_{ft}$ assigns to $U/T$ the base change $X_U \to U$. We will explain how to construct a quasi-inverse. \medskip\noindent If $T$ is a scheme, then there is a quasi-inverse by the $2$-Yoneda lemma, see Categories, Lemma \ref{categories-lemma-yoneda-2category}. Let $p : U \to T$ be a surjective \'etale morphism where $U$ is a scheme. Let $R = U \times_T U$ with projections $s, t : R \to U$. Observe that we obtain morphisms $$ \xymatrix{ \mathcal{S}_{U \times_T U \times_T U} \ar@<2ex>[r] \ar[r] \ar@<-2ex>[r] & \mathcal{S}_R \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{S}_U \ar[r] & \mathcal{S}_T } $$ satisfying various compatibilities (on the nose). \medskip\noindent Let $G : \mathcal{S}_T \to \Spacesstack'_{ft}$ be a functor over $\Sch_{fppf}$. The restriction of $G$ to $\mathcal{S}_U$ via the map displayed above corresponds to a finite type morphism $X_U \to U$ of algebraic spaces via the $2$-Yoneda lemma. Since $p \circ s = p \circ t$ we see that $R \times_{s, U} X_U$ and $R \times_{t, U} X_U$ both correspond to the restriction of $G$ to $\mathcal{S}_R$. Thus we obtain a canonical isomorphism $\varphi : X_U \times_{U, t} R \to R \times_{s, U} X_U$ over $R$. This isomorphism satisfies the cocycle condition by the various compatibilities of the diagram given above. Thus a descent datum which is effective by Bootstrap, Lemma \ref{bootstrap-lemma-descend-algebraic-space} part (2). In other words, we obtain an object $X \to T$ of the right hand side category. We omit checking the construction $G \leadsto X$ is functorial and that it is quasi-inverse to the other construction. In the case of $\Spacesstack'_{fp, flat, proper}$ one additionally uses Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-finite-presentation}, \ref{spaces-descent-lemma-descending-property-flat}, and \ref{spaces-descent-lemma-descending-property-proper} in the last step to see that $X \to T$ is of finite presentation, flat, and proper.