hcju/stacksqa huggingface dataset:
The last part is immediate from the first part and Lemma \ref{lemma-monomorphism}. Pick a field $k$ and a morphism $\Spec(k) \to \mathcal{Z}$ which is surjective, flat, and locally of finite type. Set $U = \Spec(k)$ and $R = U \times_\mathcal{Z} U$. The projections $s, t : R \to U$ are locally of finite type. Since $U$ is the spectrum of a field, it follows that $s, t$ are flat and locally of finite presentation (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}). We see that $\mathcal{Z}' = [U/R]$ is an algebraic stack by Criteria for Representability, Theorem \ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}. By Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack} we obtain a canonical morphism $$ f : \mathcal{Z}' \longrightarrow \mathcal{Z} $$ which is fully faithful. Hence this morphism is representable by algebraic spaces, see Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-algebraic-spaces} and a monomorphism, see Lemma \ref{lemma-monomorphism}. By Criteria for Representability, Lemma \ref{criteria-lemma-flat-quotient-flat-presentation} the morphism $U \to \mathcal{Z}'$ is surjective, flat, and locally of finite presentation. Hence $\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent conditions of Lemma \ref{lemma-unique-point-better}. By Algebraic Stacks, Lemma \ref{algebraic-lemma-equivalent} we may replace $\mathcal{Z}'$ by its essential image in $\mathcal{Z}$. Hence we have proved all the assertions of the lemma except for the uniqueness of $\mathcal{Z}' \subset \mathcal{Z}$. Suppose that $\mathcal{Z}'' \subset \mathcal{Z}$ is a second such algebraic stack. Then the projections $$ \mathcal{Z}' \longleftarrow \mathcal{Z}' \times_\mathcal{Z} \mathcal{Z}'' \longrightarrow \mathcal{Z}'' $$ are monomorphisms. The algebraic stack in the middle is nonempty by Lemma \ref{lemma-points-cartesian}. Hence the two projections are isomorphisms by Lemma \ref{lemma-monomorphism-into-point} and we win.
If $\mathcal{Z} \to \mathcal{X}$ is as in (1), then $\mathcal{Z}_{red} \to \mathcal{X}$ is as in (2). (See Section \ref{section-reduced} for the notion of the reduction of an algebraic stack.) Hence (1) implies (2). It is immediate that (2) implies (1). The equivalence of (2) and (3) is immediate from Lemma \ref{lemma-unique-point}. \medskip\noindent At this point we've seen the equivalence of (1) -- (3). Pick a monomorphism $f : \mathcal{Z} \to \mathcal{X}$ as in (2). Note that this implies that $f$ is fully faithful, see Lemma \ref{lemma-monomorphism}. Denote $\mathcal{Z}' \subset \mathcal{X}$ the essential image of the functor $f$. Then $f : \mathcal{Z} \to \mathcal{Z}'$ is an equivalence and hence $\mathcal{Z}'$ is an algebraic stack, see Algebraic Stacks, Lemma \ref{algebraic-lemma-equivalent}. Apply Lemma \ref{lemma-improve-unique-point} to get a strictly full subcategory $\mathcal{Z}_x \subset \mathcal{Z}'$ as in the statement of the lemma. This proves all the statements of the lemma except for uniqueness. \medskip\noindent In order to prove the uniqueness suppose that $\mathcal{Z}_x \subset \mathcal{X}$ and $\mathcal{Z}'_x \subset \mathcal{X}$ are two strictly full subcategories as in the statement of the lemma. Then the projections $$ \mathcal{Z}'_x \longleftarrow \mathcal{Z}'_x \times_\mathcal{X} \mathcal{Z}_x \longrightarrow \mathcal{Z}_x $$ are monomorphisms. The algebraic stack in the middle is nonempty by Lemma \ref{lemma-points-cartesian}. Hence the two projections are isomorphisms by Lemma \ref{lemma-monomorphism-into-point} and we win.
Let $W \to \mathcal{Z}$ be a surjective smooth morphism where $W$ is a scheme. Let $k$ be a field and let $\Spec(k) \to \mathcal{Z}$ be surjective, flat, and locally of finite presentation (see Lemma \ref{lemma-unique-point-better}). The algebraic space $T = W \times_\mathcal{Z} \Spec(k)$ is smooth over $k$ in particular regular, see Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-smooth-regular}. Since $T \to W$ is locally of finite presentation, flat, and surjective it follows that $W$ is regular, see Descent on Spaces, Lemma \ref{spaces-descent-lemma-descend-regular}. By definition this means that $\mathcal{Z}$ is regular.
Choose a field $k$ and a surjective flat locally finite presentation morphism $\Spec(k) \to \mathcal{Z}_x$. Set $T = \Spec(K) \times_\mathcal{X} \mathcal{Z}_x$. By Lemma \ref{lemma-points-cartesian} we see that $T$ is nonempty. As $\mathcal{Z}_x \to \mathcal{X}$ is a monomorphism we see that $T \to \Spec(K)$ is a monomorphism. Hence by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-monomorphism-toward-field} we see that $T = \Spec(K)$ which proves the lemma.
Let $\mathcal{Z}_x \subset \mathcal{X}$ be the full subcategory corresponding to the essential image of the functor $\mathcal{Z} \to \mathcal{X}$. Then $\mathcal{Z} \to \mathcal{Z}_x$ is an equivalence, hence $\mathcal{Z}_x$ is an algebraic stack, see Algebraic Stacks, Lemma \ref{algebraic-lemma-equivalent}. Since $\mathcal{Z}_x$ inherits all the properties of $\mathcal{Z}$ from this equivalence it is clear from the uniqueness in Lemma \ref{lemma-residual-gerbe} that $\mathcal{Z}_x$ is the residual gerbe of $\mathcal{X}$ at $x$.
Choose a field $k$ and a surjective, flat, locally finitely presented morphism $\Spec(k) \to \mathcal{Z}_x$. The morphism $\Spec(k) \to \mathcal{Y}$ factors through $\mathcal{Z}_y$ by Lemma \ref{lemma-residual-gerbe-points}. Thus $\mathcal{Z}_x \times_\mathcal{Y} \mathcal{Z}_y$ is a nonempty substack of $\mathcal{Z}_x$ hence equal to $\mathcal{Z}_x$ by Lemma \ref{lemma-monomorphism-into-point}.
Let $k'/k$ be an extension of fields. Then $$ \Spec(k') \times_\mathcal{X} \Spec(k') \longrightarrow \Spec(k') \times_\mathcal{Y} \Spec(k') $$ is the base change of the morphism in the lemma by the faithfully flat morphism $\Spec(k' \otimes k') \to \Spec(k \otimes k)$. Thus the property described in the lemma is independent of the choice of the morphism $\Spec(k) \to \mathcal{X}$ in the equivalence class of $x$. Thus we may assume that $\Spec(k) \to \mathcal{Z}_x$ is surjective, flat, and locally of finite presentation. In this situation we have $$ \mathcal{Z}_x = [\Spec(k)/R] $$ with $R = \Spec(k) \times_\mathcal{X} \Spec(k)$. See proof of Lemma \ref{lemma-improve-unique-point}. Since also $R = \Spec(k) \times_\mathcal{Y} \Spec(k)$ we conclude that the morphism $\mathcal{Z}_x \to \mathcal{Z}_y$ of Lemma \ref{lemma-residual-gerbe-functorial} is fully faithful by Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}. We conclude for example by Lemma \ref{lemma-residual-gerbe-unique}.
Since $R \to U$ is smooth, the scheme $R_u$ is smooth over $\kappa(u)$ and hence has finite dimension. On the other hand, the scheme $U$ is locally Noetherian, but this does not guarantee that $\dim_u(U)$ is finite. Thus the difference is an element of $\mathbf{Z} \cup \{\infty\}$. \medskip\noindent Let $[U'/R'] \to \mathcal{X}$ and $u' \in U'$ be a second presentation where $U'$ is a scheme and $u'$ maps to $x$. Consider the algebraic space $P = U \times_\mathcal{X} U'$. By Lemma \ref{lemma-points-cartesian} there exists a $p \in |P|$ mapping to $u$ and $u'$. Since $P \to U$ and $P \to U'$ are smooth we see that $\dim_p(P) = \dim_u(U) + \dim_p(P_u)$ and $\dim_p(P) = \dim_{u'}(U') + \dim_p(P_{u'})$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-smoothness-dimension-spaces}. Note that $$ R'_{u'} = \Spec(\kappa(u')) \times_\mathcal{X} U' \quad\text{and}\quad P_u = \Spec(\kappa(u)) \times_\mathcal{X} U' $$ Let us represent $p \in |P|$ by a morphism $\Spec(\Omega) \to P$. Since $p$ maps to both $u$ and $u'$ it induces a $2$-morphism between the compositions $\Spec(\Omega) \to \Spec(\kappa(u')) \to \mathcal{X}$ and $\Spec(\Omega) \to \Spec(\kappa(u)) \to \mathcal{X}$ which in turn defines an isomorphism $$ \Spec(\Omega) \times_{\Spec(\kappa(u'))} R'_{u'} \cong \Spec(\Omega) \times_{\Spec(\kappa(u))} P_u $$ as algebraic spaces over $\Spec(\Omega)$ mapping the $\Omega$-rational point $(1, e'(u'))$ to $(1, p)$ (some details omitted). We conclude that $$ \dim_{e'(u')}(R'_{u'}) = \dim_p(P_u) $$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}. By symmetry we have $\dim_{e(u)}(R_u) = \dim_p(P_{u'})$. Putting everything together we obtain the independence of choices.
Let $\Spec(k) \to \mathcal{X}$ be in the equivalence class of $x$. Let $k'/k$ be a field extension. Then we have to show that $\Spec(k) \to \mathcal{X}$ is quasi-compact if and only if $\Spec(k') \to \mathcal{X}$ is quasi-compact. This follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-surjection-from-quasi-compact} and the principle of Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-transformations-property-implication}.
Let $g : V'/U \to V/U$ be as above and similarly $g' : V''/U \to V'/U$ be morphisms in $\mathcal{C}/U$. So $f' = f \circ g$ and $f'' = f' \circ g' = f \circ g \circ g'$. Let $\phi \in \Mor_{\mathcal{S}_V}(f^\ast x, f^\ast y)$. Then we have \begin{eqnarray*} & & (\alpha_{g \circ g', f})_y^{-1} \circ (g \circ g')^\ast \phi \circ (\alpha_{g \circ g', f})_x \\ & = & (\alpha_{g \circ g', f})_y^{-1} \circ (\alpha_{g', g})_{f^*y}^{-1} \circ (g')^*g^\ast \phi \circ (\alpha_{g', g})_{f^*x} \circ (\alpha_{g \circ g', f})_x \\ & = & (\alpha_{g', f'})_y^{-1} \circ (g')^*(\alpha_{g, f})_y^{-1} \circ (g')^* g^\ast \phi \circ (g')^*(\alpha_{g, f})_x \circ (\alpha_{g', f'})_x \\ & = & (\alpha_{g', f'})_y^{-1} \circ (g')^*\Big( (\alpha_{g, f})_y^{-1} \circ g^\ast \phi \circ (\alpha_{g, f})_x \Big) \circ (\alpha_{g', f'})_x \end{eqnarray*} which is what we want, namely $\phi|_{V''} = (\phi|_{V'})|_{V''}$. The first equality holds because $\alpha_{g', g}$ is a transformation of functors, and hence $$ \xymatrix{ (g \circ g')^*f^*x \ar[rr]_{(g \circ g')^\ast \phi} \ar[d]_{(\alpha_{g', g})_{f^*x}} & & (g \circ g')^*f^*y \ar[d]^{(\alpha_{g', g})_{f^*y}} \\ (g')^*g^*f^*x \ar[rr]^{(g')^*g^\ast \phi} & & (g')^*g^*f^*y } $$ commutes. The second equality holds because of property (d) of a pseudo functor since $f' = f \circ g$ (see Categories, Definition \ref{categories-definition-functor-into-2-category}). The last equality follows from the fact that $(g')^*$ is a functor.
By Categories, Definition \ref{categories-definition-fibred-categories-over-C} the functor $F$ maps strongly cartesian morphisms to strongly cartesian morphisms. Hence if $f : V \to U$ is a morphism in $\mathcal{C}$, then there are canonical isomorphisms $\alpha_V : f^*F(x) \to F(f^*x)$, $\beta_V : f^*F(y) \to F(f^*y)$ such that $f^*F(x) \to F(f^*x) \to F(x)$ is the canonical morphism $f^*F(x) \to F(x)$, and similarly for $\beta_V$. Thus we may define $$ \xymatrix{ \mathit{Mor}_{\mathcal{S}_1}(x, y)(f : V \to U) \ar@{=}[r] & \Mor_{\mathcal{S}_{1, V}}(f^\ast x, f^\ast y) \ar[d] \\ \mathit{Mor}_{\mathcal{S}_2}(F(x), F(y))(f : V \to U) \ar@{=}[r] & \Mor_{\mathcal{S}_{2, V}}(f^\ast F(x), f^\ast F(y)) } $$ by $\phi \mapsto \beta_V^{-1} \circ F(\phi) \circ \alpha_V$. We omit the verification that this is compatible with the restriction mappings.
Condition (1) guarantees we have enough fibre products so that the statement makes sense. We will show that the functor $\mathcal{S}_U \to DD(\mathcal{U})$ is essentially surjective. Suppose given a descent datum $(X_i, \varphi_{ii'})$ relative to $\mathcal{U}$. By Lemma \ref{lemma-pullback} we can pull this back to a descent datum $(X_j, \varphi_{jj'})$ for $\mathcal{V}$. By assumption (2) this descent datum is effective, hence we get an object $X$ of $\mathcal{S}_U$ such that the pullback of the trivial descent datum $(X, \text{id}_X)$ by the morphism $\mathcal{V} \to \{U \to U\}$ is isomorphic to $(X_j, \varphi_{jj'})$. Next, observe that we have a diagram $$ \xymatrix{ \mathcal{V}_i \ar[r] \ar[d] & \mathcal{V} \ar[r] & \mathcal{U} \ar[d] \\ \{U_i \to U_i\} \ar[rr] \ar[rru] & & \{U \to U\} } $$ of morphisms of families of maps with fixed target of $\mathcal{C}$. This diagram does not commute, but by Lemma \ref{lemma-pullback} the pullback functors on descent data one gets are canonically isomorphic. Hence $(X, \text{id}_X)$ and $(X_i, \text{id}_{X_i})$ pull back to isomorphic objects in $DD(\mathcal{V}_i)$. Hence by assumption (3) we obtain an isomorphism $(U_i \to U)^*X \to X_i$ in the category $\mathcal{S}_{U_i}$. We omit the verification that these arrows are compatible with the morphisms $\varphi_{ii'}$; hint: use the faithfulness of the functors in condition (4). We also omit the verification that the functor $\mathcal{S}_U \to DD(\mathcal{U})$ is fully faithful.
Let $F : \mathcal{S}_1 \to \mathcal{S}_2$, $G : \mathcal{S}_2 \to \mathcal{S}_1$ be functors over $\mathcal{C}$, and let $i : F \circ G \to \text{id}_{\mathcal{S}_2}$, $j : G \circ F \to \text{id}_{\mathcal{S}_1}$ be isomorphisms of functors over $\mathcal{C}$. By Categories, Lemma \ref{categories-lemma-fibred-equivalent} we see that $\mathcal{S}_1$ is fibred if and only if $\mathcal{S}_2$ is fibred over $\mathcal{C}$. Hence we may assume that both $\mathcal{S}_1$ and $\mathcal{S}_2$ are fibred. Moreover, the proof of Categories, Lemma \ref{categories-lemma-fibred-equivalent} shows that $F$ and $G$ map strongly cartesian morphisms to strongly cartesian morphisms, i.e., $F$ and $G$ are $1$-morphisms of fibred categories over $\mathcal{C}$. This means that given $U \in \Ob(\mathcal{C})$, and $x, y \in \mathcal{S}_{1, U}$ then the presheaves $$ \mathit{Mor}_{\mathcal{S}_1}(x, y), \mathit{Mor}_{\mathcal{S}_1}(F(x), F(y)) : (\mathcal{C}/U)^{opp} \longrightarrow \textit{Sets}. $$ are identified, see Lemma \ref{lemma-presheaf-mor-map-fibred-categories}. Hence the first is a sheaf if and only if the second is a sheaf. Finally, we have to show that if every descent datum in $\mathcal{S}_1$ is effective, then so is every descent datum in $\mathcal{S}_2$. To do this, let $(X_i, \varphi_{ii'})$ be a descent datum in $\mathcal{S}_2$ relative the covering $\{U_i \to U\}$ of the site $\mathcal{C}$. Then $(G(X_i), G(\varphi_{ii'}))$ is a descent datum in $\mathcal{S}_1$ relative the covering $\{U_i \to U\}$. Let $X$ be an object of $\mathcal{S}_{1, U}$ such that the descent datum $(f_i^*X, can)$ is isomorphic to $(G(X_i), G(\varphi_{ii'}))$. Then $F(X)$ is an object of $\mathcal{S}_{2, U}$ such that the descent datum $(f_i^*F(X), can)$ is isomorphic to $(F(G(X_i)), F(G(\varphi_{ii'})))$ which in turn is isomorphic to the original descent datum $(X_i, \varphi_{ii'})$ using $i$.
Let $f : \mathcal{X} \to \mathcal{S}$ and $g : \mathcal{Y} \to \mathcal{S}$ be $1$-morphisms of stacks over $\mathcal{C}$ as defined above. The category $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C} is a fibred category according to Categories, Lemma \ref{categories-lemma-2-product-fibred-categories-over-C}. (This is where we use that $f$ and $g$ preserve strongly cartesian morphisms.) It remains to show that the morphism presheaves are sheaves and that descent relative to coverings of $\mathcal{C}$ is effective. \medskip\noindent Recall that an object of $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is given by a quadruple $(U, x, y, \phi)$. It lies over the object $U$ of $\mathcal{C}$. Next, let $(U, x', y', \phi')$ be second object lying over $U$. Recall that $\phi : f(x) \to g(y)$, and $\phi' : f(x') \to g(y')$ are isomorphisms in the category $\mathcal{S}_U$. Let us use these isomorphisms to identify $z = f(x) = g(y)$ and $z' = f(x') = g(y')$. With this identifications it is clear that $$ \mathit{Mor}((U, x, y, \phi), (U, x', y', \phi')) = \mathit{Mor}(x, x') \times_{\mathit{Mor}(z, z')} \mathit{Mor}(y, y') $$ as presheaves. However, as the fibred product in the category of presheaves preserves sheaves (Sites, Lemma \ref{sites-lemma-limit-sheaf}) we see that this is a sheaf. \medskip\noindent Let $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a covering of the site $\mathcal{C}$. Let $(X_i, \chi_{ij})$ be a descent datum in $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ relative to $\mathcal{U}$. Write $X_i = (U_i, x_i, y_i, \phi_i)$ as above. Write $\chi_{ij} = (\varphi_{ij}, \psi_{ij})$ as in the definition of the category $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ (see Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}). It is clear that $(x_i, \varphi_{ij})$ is a descent datum in $\mathcal{X}$ and that $(y_i, \psi_{ij})$ is a descent datum in $\mathcal{Y}$. Since $\mathcal{X}$ and $\mathcal{Y}$ are stacks these descent data are effective. Thus we get $x \in \Ob(\mathcal{X}_U)$, and $y \in \Ob(\mathcal{Y}_U)$ with $x_i = x|_{U_i}$, and $y_i = y|_{U_i}$ compatibly with descent data. Set $z = f(x)$ and $z' = g(y)$ which are both objects of $\mathcal{S}_U$. The morphisms $\phi_i$ are elements of $\mathit{Isom}(z, z')(U_i)$ with the property that $\phi_i|_{U_i \times_U U_j} = \phi_j|_{U_i \times_U U_j}$. Hence by the sheaf property of $\mathit{Isom}(z, z')$ we obtain an isomorphism $\phi : z = f(x) \to z' = g(y)$. We omit the verification that the canonical descent datum associated to the object $(U, x, y, \phi)$ of $(\mathcal{X} \times_\mathcal{S} \mathcal{Y})_U$ is isomorphic to the descent datum we started with.
Recall that the morphisms in $\mathcal{S}'$ are exactly the strongly cartesian morphisms of $\mathcal{S}$, and that any isomorphism of $\mathcal{S}$ is such a morphism. Hence descent data in $\mathcal{S}'$ are exactly the same thing as descent data in $\mathcal{S}$. Now apply Lemma \ref{lemma-stack-equivalences}. Some details omitted.
By Categories, Lemma \ref{categories-lemma-setoid-fibres} we see that a category $\mathcal{S}$ over $\mathcal{C}$ is fibred in setoids over $\mathcal{C}$ if and only if it is equivalent over $\mathcal{C}$ to a category fibred in sets. Hence we see that $\mathcal{S}_1$ is fibred in setoids over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is fibred in setoids over $\mathcal{C}$. Hence now the lemma follows from Lemma \ref{lemma-stack-in-setoids-characterize}.
We may assume that $\mathcal{T}_2$ is the category $\mathcal{S}_2 \times_{\mathcal{S}_1} \mathcal{T}_1$ described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. By Categories, Lemma \ref{categories-lemma-equivalence-fibred-categories} the faithfulness of $G, G'$ can be checked on fibre categories. Suppose that $y, y'$ are objects of $\mathcal{T}_1$ over the object $U$ of $\mathcal{C}$. Let $\alpha, \beta : y \to y'$ be morphisms of $(\mathcal{T}_1)_U$ such that $G(\alpha) = G(\beta)$. Our object is to show that $\alpha = \beta$. Considering instead $\gamma = \alpha^{-1} \circ \beta$ we see that $G(\gamma) = \text{id}_{G(y)}$ and we have to show that $\gamma = \text{id}_y$. By assumption we can find a covering $\{U_i \to U\}$ such that $G(y)|_{U_i}$ is in the essential image of $F :(\mathcal{S}_2)_{U_i} \to (\mathcal{S}_1)_{U_i}$. Since it suffices to show that $\gamma|_{U_i} = \text{id}$ for each $i$, we may therefore assume that we have $f : F(x) \to G(y)$ for some object $x$ of $\mathcal{S}_2$ over $U$ and morphisms $f$ of $(\mathcal{S}_1)_U$. In this case we get a morphism $$ (1, \gamma) : (U, x, y, f) \longrightarrow (U, x, y, f) $$ in the fibre category of $\mathcal{S}_2 \times_{\mathcal{S}_1} \mathcal{T}_1$ over $U$ whose image under $G'$ in $\mathcal{S}_1$ is $\text{id}_x$. As $G'$ is faithful we conclude that $\gamma = \text{id}_y$ and we win.
We may assume that $\mathcal{T}_2$ is the category $\mathcal{S}_2 \times_{\mathcal{S}_1} \mathcal{T}_1$ described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. Pick $U \in \Ob(\mathcal{C})$ and $y \in \Ob((\mathcal{T}_1)_U)$. We have to show that the sheaf $\mathit{Aut}(y)$ on $\mathcal{C}/U$ is trivial. To to this we may replace $U$ by the members of a covering of $U$. Hence by assumption (2) we may assume that there exists an object $x \in \Ob((\mathcal{S}_2)_U)$ and an isomorphism $f : F(x) \to G(y)$. Then $y' = (U, x, y, f)$ is an object of $\mathcal{T}_2$ over $U$ which is mapped to $y$ under the projection $\mathcal{T}_2 \to \mathcal{T}_1$. Because $F$ is fully faithful by (1) the map $\mathit{Aut}(y') \to \mathit{Aut}(y)$ is surjective, use the explicit description of morphisms in $\mathcal{T}_2$ in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. Since by (3) the sheaf $\mathit{Aut}(y')$ is trivial we get the result of the lemma.
Omitted. Hint: Suppose that $x' \in \Ob(\mathcal{S}'_U)$. By the result of Lemma \ref{lemma-stackify} there exists a covering $\{U_i \to U\}_{i \in I}$ such that $x'|_{U_i} = G(x_i)$ for some $x_i \in \Ob(\mathcal{S}_{U_i})$. Moreover, there exist coverings $\{U_{ijk} \to U_i \times_U U_j\}$ and isomorphisms $\alpha_{ijk} : x_i|_{U_{ijk}} \to x_j|_{U_{ijk}}$ with $G(\alpha_{ijk}) = \text{id}_{x'|_{U_{ijk}}}$. Set $y_i = F(x_i)$. Then you can check that $$ F(\alpha_{ijk}) : y_i|_{U_{ijk}} \to y_j|_{U_{ijk}} $$ agree on overlaps and therefore (as $\mathcal{X}$ is a stack) define a morphism $\beta_{ij} : y_i|_{U_i \times_U U_j} \to y_j|_{U_i \times_U U_j}$. Next, you check that the $\beta_{ij}$ define a descent datum. Since $\mathcal{X}$ is a stack these descent data are effective and we find an object $y$ of $\mathcal{X}_U$ agreeing with $G(x_i)$ over $U_i$. The hint is to set $H(x') = y$.
Let us denote $\mathcal{X}', \mathcal{Y}', \mathcal{Z}'$ the stackifications and $\mathcal{W}$ the stackification of $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$. By construction of $2$-fibre products there is a canonical $1$-morphism $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{Z}'$. As the second $2$-fibre product is a stack (see Lemma \ref{lemma-2-product-stacks}) this $1$-morphism induces a $1$-morphism $h : \mathcal{W} \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{Z}'$ by the universal property of stackification, see Lemma \ref{lemma-stackify-universal-property}. Now $h$ is a morphism of stacks, and we may check that it is an equivalence using Lemmas \ref{lemma-characterize-ff} and \ref{lemma-characterize-essentially-surjective-when-ff}. \medskip\noindent Thus we first prove that $h$ induces isomorphisms of $\mathit{Mor}$-sheaves. Let $\xi, \xi'$ be objects of $\mathcal{W}$ over $U \in \Ob(\mathcal{C})$. We want to show that $$ h : \mathit{Mor}(\xi, \xi') \longrightarrow \mathit{Mor}(h(\xi), h(\xi')) $$ is an isomorphism. To do this we may work locally on $U$ (see Sites, Section \ref{sites-section-glueing-sheaves}). Hence by construction of $\mathcal{W}$ (see Lemma \ref{lemma-stackify}) we may assume that $\xi, \xi'$ actually come from objects $(x, z, \alpha)$ and $(x', z', \alpha')$ of $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $U$. By the same lemma once more we see that in this case $\mathit{Mor}(\xi, \xi')$ is the sheafification of $$ V/U \longmapsto \Mor_{\mathcal{X}_V}(x|_V, x'|_V) \times_{\Mor_{\mathcal{Y}_V}(f(x)|_V, f(x')|_V)} \Mor_{\mathcal{Z}_V}(z|_V, z'|_V) $$ and that $\mathit{Mor}(h(\xi), h(\xi'))$ is equal to the fibre product $$ \mathit{Mor}(i(x), i(x')) \times_{\mathit{Mor}(j(f(x)), j(f(x'))} \mathit{Mor}(k(z), k(z')) $$ where $i : \mathcal{X} \to \mathcal{X}'$, $j : \mathcal{Y} \to \mathcal{Y}'$, and $k : \mathcal{Z} \to \mathcal{Z}'$ are the canonical functors. Thus the first displayed map of this paragraph is an isomorphism as sheafification is exact (and hence the sheafification of a fibre product of presheaves is the fibre product of the sheafifications). \medskip\noindent Finally, we have to check that any object of $\mathcal{X}' \times_{\mathcal{Y}'} \mathcal{Z}'$ over $U$ is locally on $U$ in the essential image of $h$. Write such an object as a triple $(x', z', \alpha)$. Then $x'$ locally comes from an object of $\mathcal{X}$, $z'$ locally comes from an object of $\mathcal{Z}$, and having made suitable replacements for $x'$, $z'$ the morphism $\alpha$ of $\mathcal{Y}'_U$ locally comes from a morphism of $\mathcal{Y}$. In other words, we have shown that any object of $\mathcal{X}' \times_{\mathcal{Y}'} \mathcal{Z}'$ over $U$ is locally on $U$ in the essential image of $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{Z}'$, hence a fortiori it is locally in the essential image of $h$.