hcju/stacksqa huggingface dataset:
This is just a reformulation of a special case of the theorem on formal functions, Theorem \ref{theorem-formal-functions}. Let us spell it out. Note that $\mathcal{O}_{Y, \overline{y}}$ is a Noetherian local ring, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-Noetherian-local-ring-Noetherian}. Consider the canonical morphism $c : \Spec(\mathcal{O}_{Y, \overline{y}}) \to Y$. This is a flat morphism as it identifies local rings. Denote $f' : X' \to \Spec(\mathcal{O}_{Y, \overline{y}})$ the base change of $f$ to this local ring. We see that $c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'$ by Lemma \ref{lemma-flat-base-change-cohomology}. Moreover, we have canonical identifications $X_n = X'_n$ for all $n \geq 1$. \medskip\noindent Hence we may assume that $Y = \Spec(A)$ is the spectrum of a strictly henselian Noetherian local ring $A$ with maximal ideal $\mathfrak m$ and that $\overline{y} \to Y$ is equal to $\Spec(A/\mathfrak m) \to Y$. It follows that $$ \left(R^pf_*\mathcal{F}\right)_{\overline{y}} = \Gamma(Y, R^pf_*\mathcal{F}) = H^p(X, \mathcal{F}) $$ because $(Y, \overline{y})$ is an initial object in the category of \'etale neighbourhoods of $\overline{y}$. The morphisms $c_n$ are each closed immersions. Hence their base changes $i_n$ are closed immersions as well. Note that $i_{n, *}\mathcal{F}_n = i_{n, *}i_n^*\mathcal{F} = \mathcal{F}/\mathfrak m^n\mathcal{F}$. By the Leray spectral sequence for $i_n$, and Lemma \ref{lemma-finite-pushforward-coherent} we see that $$ H^p(X_n, \mathcal{F}_n) = H^p(X, i_{n, *}\mathcal{F}) = H^p(X, \mathcal{F}/\mathfrak m^n\mathcal{F}) $$ Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win.
Let $\kappa(\overline{y})$ be the residue field of the local ring of $\mathcal{O}_{Y, \overline{y}}$. As in Lemma \ref{lemma-formal-functions-stalk} we set $X_{\overline{y}} = X_1 = \Spec(\kappa(\overline{y})) \times_Y X$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-finite-at-point} the morphism $f : X \to Y$ is quasi-finite at each of the points of the fibre of $X \to Y$ over $\overline{y}$. It follows that $X_{\overline{y}} \to \overline{y}$ is separated and quasi-finite. Hence $X_{\overline{y}}$ is a scheme by Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}. Since it is quasi-compact its underlying topological space is a finite discrete space. Then it is an affine scheme by Schemes, Lemma \ref{schemes-lemma-scheme-finite-discrete-affine}. By Lemma \ref{lemma-image-affine-finite-morphism-affine-Noetherian} it follows that the algebraic spaces $X_n$ are affine schemes as well. Moreover, the underlying topological of each $X_n$ is the same as that of $X_1$. Hence it follows that $H^p(X_n, \mathcal{F}_n) = 0$ for all $p > 0$. Hence we see that $(R^pf_*\mathcal{F})_{\overline{y}}^\wedge = 0$ by Lemma \ref{lemma-formal-functions-stalk}. Note that $R^pf_*\mathcal{F}$ is coherent by Lemma \ref{lemma-proper-pushforward-coherent} and hence $R^pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma \ref{algebra-lemma-completion-tensor} this implies that $(R^pf_*\mathcal{F})_{\overline{y}} = 0$.
Let $\kappa(\overline{y})$ be the residue field of the local ring of $\mathcal{O}_{Y, \overline{y}}$. As in Lemma \ref{lemma-formal-functions-stalk} we set $X_{\overline{y}} = X_1 = \Spec(\kappa(\overline{y})) \times_Y X$. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_n$ is the same as that of $X_{\overline{y}}$. Hence $H^p(X_n, \mathcal{F}_n) = 0$ for all $p > d$ by Lemma \ref{lemma-vanishing-above-dimension}. Hence we see that $(R^pf_*\mathcal{F})_{\overline{y}}^\wedge = 0$ by Lemma \ref{lemma-formal-functions-stalk} for $p > d$. Note that $R^pf_*\mathcal{F}$ is coherent by Lemma \ref{lemma-proper-pushforward-coherent} and hence $R^pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma \ref{algebra-lemma-completion-tensor} this implies that $(R^pf_*\mathcal{F})_{\overline{y}} = 0$.
A finite morphism is proper according to Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-proper}. A finite morphism is quasi-finite according to Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-quasi-finite}. A quasi-finite morphism has discrete fibres $X_k$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite}. Hence a finite morphism is proper and has discrete fibres $X_k$. \medskip\noindent Assume $f$ is proper with discrete fibres $X_k$. We want to show $f$ is finite. In fact it suffices to prove $f$ is affine. Namely, if $f$ is affine, then it follows that $f$ is integral by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-universally-closed} whereupon it follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-integral} that $f$ is finite. \medskip\noindent To show that $f$ is affine we may assume that $Y$ is affine, and our goal is to show that $X$ is affine too. Since $f$ is proper we see that $X$ is separated and quasi-compact. We will show that for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. This implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ by Lemmas \ref{lemma-directed-colimit-coherent} and \ref{lemma-colimits}. Then it follows that $X$ is affine from Proposition \ref{proposition-vanishing-affine}. By Lemma \ref{lemma-higher-direct-images-zero-finite-fibre} we conclude that the stalks of $R^1f_*\mathcal{F}$ are zero for all geometric points of $Y$. In other words, $R^1f_*\mathcal{F} = 0$. Hence we see from the Leray Spectral Sequence for $f$ that $H^1(X , \mathcal{F}) = H^1(Y, f_*\mathcal{F})$. Since $Y$ is affine, and $f_*\mathcal{F}$ is quasi-coherent (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}) we conclude $H^1(Y, f_*\mathcal{F}) = 0$ from Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. Hence $H^1(X, \mathcal{F}) = 0$ as desired.
The morphism $f$ is quasi-finite at all the geometric points of $X$ lying over $\overline{y}$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-finite-at-point}. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part} the set of points at which $f$ is quasi-finite is an open subspace $U \subset X$. Let $Z = X \setminus U$. Then $\overline{y} \not \in f(Z)$. Since $f$ is proper the set $f(Z) \subset Y$ is closed. Choose any open neighbourhood $V \subset Y$ of $\overline{y}$ with $Z \cap V = \emptyset$. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence $f^{-1}(V) \to V$ has discrete fibres $X_k$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite}) which are quasi-compact hence finite. Thus $f^{-1}(V) \to V$ is finite by Lemma \ref{lemma-characterize-finite}.
We will use the spectral sequence of Lemma \ref{lemma-alternating-spectral-sequence}. The lemma applies since $f$ is separated as $U$ is separated, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-compose-after-separated}. Since $X$ is separated the scheme $U \times_X \ldots \times_X U$ is a closed subscheme of $U \times_{\Spec(\mathbf{Z})} \ldots \times_{\Spec(\mathbf{Z})} U$ hence is affine. Thus $W_p$ is affine. Hence $U_p = W_p/S_{p + 1}$ is an affine scheme by Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence}. The discussion in Section \ref{section-higher-direct-image} shows that cohomology of quasi-coherent sheaves on $W_p$ (as an algebraic space) agrees with the cohomology of the corresponding quasi-coherent sheaf on the underlying affine scheme, hence vanishes in positive degrees by Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. By Lemma \ref{lemma-quasi-coherent-twist} the sheaves $\mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)$ are quasi-coherent. Hence $H^q(W_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))$ is zero when $q > 0$. By our definition of the integer $d$ we see that $W_p = \emptyset$ for $p \geq d$. Hence also $H^0(W_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))$ is zero when $p \geq d$. This proves the proposition.
Choose an affine scheme $U = \Spec(B)$ and a surjective \'etale morphism $\varphi : U \to X$. Set $R = U \times_X U$. As $p$ is separated (Lemma \ref{lemma-vanishing-separated}) we see that $R$ is a closed subscheme of $U \times_{\Spec(A)} U = \Spec(B \otimes_A B)$. Hence $R = \Spec(C)$ is affine too and the ring map $$ B \otimes_A B \longrightarrow C $$ is surjective. Let us denote the two maps $s, t : B \to C$ as usual. Pick $g_1, \ldots, g_m \in B$ such that $s(g_1), \ldots, s(g_m)$ generate $C$ over $t : B \to C$ (which is possible as $t : B \to C$ is of finite presentation and the displayed map is surjective). Then $g_1, \ldots, g_m$ give global sections of $\varphi_*\mathcal{O}_U$ and the map $$ \mathcal{O}_X[z_1, \ldots, z_n] \longrightarrow \varphi_*\mathcal{O}_U, \quad z_j \longmapsto g_j $$ is surjective: you can check this by restricting to $U$. Namely, $\varphi^*\varphi_*\mathcal{O}_U = t_*\mathcal{O}_R$ (by Lemma \ref{lemma-flat-base-change-cohomology}) hence you get exactly the condition that $s(g_i)$ generate $C$ over $t : B \to C$. By the vanishing of $H^1$ of the kernel we see that $$ \Gamma(X, \mathcal{O}_X[x_1, \ldots, x_n]) = A[x_1, \ldots, x_n] \longrightarrow \Gamma(X, \varphi_*\mathcal{O}_U) = \Gamma(U, \mathcal{O}_U) = B $$ is surjective. Thus we conclude that $B$ is a finite type $A$-algebra. Hence $X \to \Spec(A)$ is of finite type and separated. By Lemma \ref{lemma-vanishing-injective} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite} it is also locally quasi-finite. Hence $X \to \Spec(A)$ is representable by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable} and $X$ is a scheme. Finally $X$ is affine, hence equal to $\Spec(A)$, by an application of Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-compact-h1-zero-covering}.
Let $F$ be the functor constructed using $X'$, $T'$, $W$, $g$ in this section. By Lemma \ref{lemma-functor-is-solution} it suffices to show that $F$ corresponds to an algebraic space $X$ locally of finite type over $S$. In order to do this, we will apply Proposition \ref{proposition-spaces-diagonal-representable-noetherian}. Namely, by Lemma \ref{lemma-diagonal-contractions} the diagonal of $F$ is representable by closed immersions and by Lemmas \ref{lemma-sheaf}, \ref{lemma-limit-preserving}, \ref{lemma-rs}, \ref{lemma-finite-dim}, \ref{lemma-formal-object-effective}, and \ref{lemma-openness-versality} we have axioms [0], [1], [2], [3], [4], and [5].
We have to show that $\mathcal{F}$ is (a) cofibred in groupoids over $\mathcal{C}_\Lambda$ and (b) that $\mathcal{F}(k)$ is a category equivalent to a category with a single object and a single morphism. \medskip\noindent Proof of (a). The fibre categories of $\mathcal{F}$ over $\mathcal{C}_\Lambda$ are groupoids as the fibre categories of $\mathcal{X}$ are groupoids. Let $A \to A'$ be a morphism of $\mathcal{C}_\Lambda$ and let $x_0 \to x$ be an object of $\mathcal{F}(A)$. Because $\mathcal{X}$ is fibred in groupoids, we can find a morphism $x' \to x$ lying over $\Spec(A') \to \Spec(A)$. Since the composition $A \to A' \to k$ is equal the given map $A \to k$ we see (by uniqueness of pullbacks up to isomorphism) that the pullback via $\Spec(k) \to \Spec(A')$ of $x'$ is $x_0$, i.e., that there exists a morphism $x_0 \to x'$ lying over $\Spec(k) \to \Spec(A')$ compatible with $x_0 \to x$ and $x' \to x$. This proves that $\mathcal{F}$ has pushforwards. We conclude by (the dual of) Categories, Lemma \ref{categories-lemma-fibred-groupoids}. \medskip\noindent Proof of (b). If $A = k$, then $\Spec(k) = \Spec(A)$ and since $\mathcal{X}$ is fibred in groupoids over $(\Sch/S)_{fppf}$ we see that given any object $x_0 \to x$ in $\mathcal{F}(k)$ the morphism $x_0 \to x$ is an isomorphism. Hence every object of $\mathcal{F}(k)$ is isomorphic to $x_0 \to x_0$. Clearly the only self morphism of $x_0 \to x_0$ in $\mathcal{F}$ is the identity.
Case (1) is a matter of unwinding the definitions. Assumption (2) implies (1) by Criteria for Representability, Lemma \ref{criteria-lemma-representable-by-spaces-formally-smooth}. Assumption (3) implies (2) by More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-smooth-formally-smooth} and the principle of Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-transformations-property-implication}.
Let $y'$ be an object of left hand side. The sheaf $\mathit{Isom}(y', y')$ on the category of schemes over $Y'$ is representable by an algebraic space $I$ over $Y'$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. We conclude that the functor of the lemma is fully faithful as $Y'$ is the pushout in the category of algebraic spaces as well as the category of schemes, see Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-pushout-along-thickening-schemes}. \medskip\noindent Let $(y, x', f)$ be an object of the right hand side. Here $f : y|_X \to x'|_X$ is an isomorphism. To finish the proof we have to construct an object $y'$ of $\mathcal{Z}_{Y'}$ whose restrictions to $Y$ and $X'$ agree with $y$ and $x'$ in a manner compatible with $\varphi$. In fact, it suffices to construct $y'$ fppf locally on $Y'$, see Stacks, Lemma \ref{stacks-lemma-characterize-essentially-surjective-when-ff}. Choose a representable algebraic stack $\mathcal{W}$ and a surjective smooth morphism $\mathcal{W} \to \mathcal{Z}$. Then $$ (\Sch/Y)_{fppf} \times_{y, \mathcal{Z}} \mathcal{W} \quad\text{and}\quad (\Sch/X')_{fppf} \times_{x', \mathcal{Z}} \mathcal{W} $$ are algebraic stacks representable by algebraic spaces $V$ and $U'$ smooth over $Y$ and $X'$. The isomorphism $f$ induces an isomorphism $\varphi : V \times_Y X \to U' \times_{X'} X$ over $X$. By Pushouts of Spaces, Lemmas \ref{spaces-pushouts-lemma-pushout-along-thickening} and \ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat} we see that the pushout $V' = V \amalg_{V \times_Y X} U'$ is an algebraic space smooth over $Y'$ whose base change to $Y$ and $X'$ recovers $V$ and $U'$ in a manner compatible with $\varphi$. \medskip\noindent Let $W$ be the algebraic space representing $\mathcal{W}$. The projections $V \to W$ and $U' \to W$ agree as morphisms over $V \times_Y X \cong U' \times_{X'} X$ hence the universal property of the pushout determines a morphism of algebraic spaces $V' \to W$. Choose a scheme $Y_1'$ and a surjective \'etale morphism $Y_1' \to V'$. Set $Y_1 = Y \times_{Y'} Y_1'$, $X_1' = X' \times_{Y'} Y_1'$, $X_1 = X \times_{Y'} Y_1'$. The composition $$ (\Sch/Y_1') \to (\Sch/V') \to (\Sch/W) = \mathcal{W} \to \mathcal{Z} $$ corresponds by the $2$-Yoneda lemma to an object $y_1'$ of $\mathcal{Z}$ over $Y_1'$ whose restriction to $Y_1$ and $X_1'$ agrees with $y|_{Y_1}$ and $x'|_{X_1'}$ in a manner compatible with $f|_{X_1}$. Thus we have constructed our desired object smooth locally over $Y'$ and we win.
This is formal. Let  $$ \xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' = Y \amalg_X X' } $$ be a diagram as in Definition \ref{definition-RS}. We have to show that $$ (\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_{Y'} \longrightarrow (\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_Y \times_{(\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_X} (\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_{X'} $$ is an equivalence. Using the definition of the $2$-fibre product this becomes \begin{equation} \label{equation-RS-fibre-product} \mathcal{X}_{Y'} \times_{\mathcal{Y}_{Y'}} \mathcal{Z}_{Y'} \longrightarrow (\mathcal{X}_Y \times_{\mathcal{Y}_Y} \mathcal{Z}_Y) \times_{(\mathcal{X}_X \times_{\mathcal{Y}_X} \mathcal{Z}_X)} (\mathcal{X}_{X'} \times_{\mathcal{Y}_{X'}} \mathcal{Z}_{X'}). \end{equation} We are given that each of the functors $$ \mathcal{X}_{Y'} \to \mathcal{X}_Y \times_{\mathcal{Y}_Y} \mathcal{Z}_Y, \quad \mathcal{Y}_{Y'} \to \mathcal{X}_X \times_{\mathcal{Y}_X} \mathcal{Z}_X, \quad \mathcal{Z}_{Y'} \to \mathcal{X}_{X'} \times_{\mathcal{Y}_{X'}} \mathcal{Z}_{X'} $$ are equivalences. An object of the right hand side of (\ref{equation-RS-fibre-product}) is a system $$ ((x_Y, z_Y, \phi_Y), (x_{X'}, z_{X'}, \phi_{X'}), (\alpha, \beta)). $$ Then $(x_Y, x_{Y'}, \alpha)$ is isomorphic to the image of an object $x_{Y'}$ in $\mathcal{X}_{Y'}$ and $(z_Y, z_{Y'}, \beta)$ is isomorphic to the image of an object $z_{Y'}$ of $\mathcal{Z}_{Y'}$. The pair of morphisms $(\phi_Y, \phi_{X'})$ corresponds to a morphism $\psi$ between the images of $x_{Y'}$ and $z_{Y'}$ in $\mathcal{Y}_{Y'}$. Then $(x_{Y'}, z_{Y'}, \psi)$ is an object of the left hand side of (\ref{equation-RS-fibre-product}) mapping to the given object of the right hand side. This proves that (\ref{equation-RS-fibre-product}) is essentially surjective. We omit the proof that it is fully faithful.
Set $\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}$. Let $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ be ring maps in $\mathcal{C}_\Lambda$ with $f_2$ surjective. We have to show that the functor $$ \mathcal{F}(A_1 \times_A A_2) \longrightarrow \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2) $$ is an equivalence, see Formal Deformation Theory, Lemma \ref{formal-defos-lemma-RS-2-categorical}. Set $X = \Spec(A)$, $X' = \Spec(A_2)$, $Y = \Spec(A_1)$ and $Y' = \Spec(A_1 \times_A A_2)$. Note that $Y' = Y \amalg_X X'$ in the category of schemes, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. We know that in the diagram of functors of fibre categories $$ \xymatrix{ \mathcal{X}_{Y'} \ar[r] \ar[d] & \mathcal{X}_Y \times_{\mathcal{X}_X} \mathcal{X}_{X'} \ar[d] \\ \mathcal{X}_{\Spec(k)} \ar@{=}[r] & \mathcal{X}_{\Spec(k)} } $$ the top horizontal arrow is an equivalence by Definition \ref{definition-RS}. Since $\mathcal{F}(B)$ is the category of objects of $\mathcal{X}_{\Spec(B)}$ with an identification with $x_0$ over $k$ we win.
Consider a factorization $$ \Spec(l) \to \Spec(B) \to S $$ as in (\ref{equation-factor}). By definition we have $$ (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k}(B) = \mathcal{F}_{\mathcal{X}, k, x_0}(B \times_l k) $$ see Formal Deformation Theory, Situation \ref{formal-defos-situation-change-of-fields}. Thus an object of this is a morphism $x_0 \to x$ of $\mathcal{X}$ lying over the morphism $\Spec(k) \to \Spec(B \times_l k)$. Choosing pullback functor for $\mathcal{X}$ we can associate to $x_0 \to x$ the morphism $x_{l, 0} \to x_B$ where $x_B$ is the restriction of $x$ to $\Spec(B)$ (via the morphism $\Spec(B) \to \Spec(B \times_l k)$ coming from $B \times_l k \subset B$). This construction is functorial in $B$ and compatible with morphisms. \medskip\noindent Next, assume $\mathcal{X}$ satisfies (RS). Consider the diagrams $$ \vcenter{ \xymatrix{ l & B \ar[l] \\ k \ar[u] & B \times_l k \ar[l] \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ \Spec(l) \ar[d] \ar[r] & \Spec(B) \ar[d] \\ \Spec(k) \ar[r] & \Spec(B \times_l k) } } $$ The diagram on the left is a fibre product of rings. The diagram on the right is a pushout in the category of schemes, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. These schemes are all of finite type over $S$ (see remarks following Definition \ref{definition-RS}). Hence (RS) kicks in to give an equivalence of fibre categories $$ \mathcal{X}_{\Spec(B \times_l k)} \longrightarrow \mathcal{X}_{\Spec(k)} \times_{\mathcal{X}_{\Spec(l)}} \mathcal{X}_{\Spec(B)} $$ This implies that the functor defined above gives an equivalence of fibre categories. Hence the functor is an equivalence on categories cofibred in groupoids by (the dual of) Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories}.
By Lemma \ref{lemma-fibre-product-RS} we see that $\mathcal{W}$ satisfies (RS) and hence the lemma makes sense. To see the lemma is true, apply Lemmas \ref{lemma-fibre-product-deformation-categories} and \ref{lemma-deformation-category} and Formal Deformation Theory, Lemma \ref{formal-defos-lemma-deformation-categories-fiber-product-morphisms}.
The left and the right hand side of (\ref{equation-approximation}) for $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ are simply the $2$-fibre products of the left and the right hand side of (\ref{equation-approximation}) for $\mathcal{X}$, $\mathcal{Z}$ over $\mathcal{Y}$. Hence the result follows as taking $2$-fibre products is compatible with equivalences of categories, see Categories, Lemma \ref{categories-lemma-equivalence-2-fibre-product}.
Under each of the three assumptions $\mathcal{X}$ is representable by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition \ref{algebraic-proposition-algebraic-stack-no-automorphisms}. It is clear that (1) and (2) are equivalent as a functor between setoids is an equivalence if and only if it is surjective on isomorphism classes. Finally, (1) and (3) are equivalent by Limits of Spaces, Proposition \ref{spaces-limits-proposition-characterize-locally-finite-presentation}.
We apply Criteria for Representability, Lemma \ref{criteria-lemma-check-property-limit-preserving}. Let $V$ be an affine scheme $V$ locally of finite presentation over $S$ and let $\theta$ be an object of $\mathcal{X} \times \mathcal{X}$ over $V$. Let $F_\theta$ be an algebraic space representing $\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, \theta} (\Sch/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism (see Algebraic Stacks, Section \ref{algebraic-section-morphisms-representable-by-algebraic-spaces}). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas \ref{lemma-fibre-product-limit-preserving} and \ref{lemma-limit-preserving-algebraic-space} we see that $F_\theta \to S$ is locally of finite presentation. It follows that $F_\theta \to V$ is locally of finite type by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}.
Observe that $\mathcal{O}_{U, u_0}$ is a Noetherian local $S$-algebra with residue field $k$. Hence $R = \mathcal{O}_{U, u_0}^\wedge$ is an object of $\mathcal{C}_\Lambda^\wedge$, see Formal Deformation Theory, Definition \ref{formal-defos-definition-completion-CLambda}. Recall that $\xi$ is versal if $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth and $x$ is versal at $u_0$ if $\hat x : \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth. There is an identification of predeformation categories $$ \underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0}, $$ see Formal Deformation Theory, Remark \ref{formal-defos-remark-formal-objects-yoneda} for notation. Namely, given an Artinian local $S$-algebra $A$ with residue field identified with $k$ we have $$ \Mor_{\mathcal{C}_\Lambda^\wedge}(R, A) = \{\varphi \in \Mor_S(\Spec(A), U) \mid \varphi|_{\Spec(k)} = u_0\} $$ Unwinding the definitions the reader verifies that the resulting map $$ \underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \xrightarrow{\hat x} \mathcal{F}_{\mathcal{X}, k, x_0}, $$ is equal to $\underline{\xi}$ and we see that the lemma is true.
Note that $(\Sch/U)_{fppf}$ satisfies (RS) by Lemma \ref{lemma-algebraic-stack-RS}. Hence the functor of the lemma is the functor $$ (\mathcal{F}_{(\Sch/U)_{fppf}, k , u_0})_{l/k} \longrightarrow (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k} $$ associated to $\hat x$, see Lemma \ref{lemma-change-of-field}. Hence the lemma follows from Formal Deformation Theory, Lemma \ref{formal-defos-lemma-change-of-fields-smooth}.