hcju/stacksqa huggingface dataset:
Denote $\mathcal{C}$ the category of compactifications and denote $Q : \mathcal{C} \to \mathcal{C}'$ the localization functor of Categories, Lemma \ref{categories-lemma-properties-right-localization}. Denote $\mathcal{D}$ the category of compactifyable schemes over $S$. It is clear from the lemma just cited and our choice of multiplicative system that we obtain a functor $\mathcal{C}' \to \mathcal{D}$. This functor is clearly essentially surjective. If $f : X \to Y$ is a morphism of compactifyable schemes, then we choose an open immersion $Y \to \overline{Y}$ into a scheme proper over $S$, and then we choose an embedding $X \to \overline{X}$ into a scheme $\overline{X}$ proper over $\overline{Y}$ (possible by Lemma \ref{lemma-compactifyable} applied to $X \to \overline{Y}$). This gives a morphism $(X, \overline{X}) \to (Y, \overline{Y})$ of compactifications which produces our given morphism $X \to Y$. Finally, suppose given a pair of morphisms in the localized category with the same source and target: say $$ a = ((f, \overline{f}) : (X', \overline{X}') \to (Y, \overline{Y}), (u, \overline{u}) : (X', \overline{X}') \to (X, \overline{X})) $$ and $$ b = ((g, \overline{g}) : (X'', \overline{X}'') \to (Y, \overline{Y}), (v, \overline{v}) : (X'', \overline{X}'') \to (X, \overline{X})) $$ which produce the same morphism $X \to Y$ over $S$, in other words $f \circ u^{-1} = g \circ v^{-1}$. By Categories, Lemma \ref{categories-lemma-morphisms-right-localization} we may assume that $(X', \overline{X}') = (X'', \overline{X}'')$ and $(u, \overline{u}) = (v, \overline{v})$. In this case we can consider the equalizer $\overline{X}''' \subset \overline{X}'$ of $\overline{f}$ and $\overline{g}$. The morphism $(w, \overline{w}) : (X', \overline{X}''') \to (X', \overline{X}')$ is in the multiplicative subset and we see that $a = b$ in the localized category by precomposing with $(w, \overline{w})$.
Proof of (1). The assumption that $Z_i \to X$ is of finite presentation signifies that the quasi-coherent ideal sheaf $\mathcal{I}_i$ of $Z_i$ is of finite type, see  Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-finite-presentation}. Denote $Z \subset X$ the closed subscheme cut out by the product $\mathcal{I}_1 \mathcal{I}_2$. Observe that $Z \cap U$ is the disjoint union of $Z_1 \cap U$ and $Z_2 \cap U$. By Divisors, Lemma \ref{divisors-lemma-separate-disjoint-opens-by-blowing-up} there is a $U \cap Z$-admissible blowup $Z' \to Z$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. Denote $Y \subset Z$ the center of this blowing up. Then $Y \to X$ is a closed immersion of finite presentation as the composition of $Y \to Z$ and $Z \to X$ (Divisors, Definition \ref{divisors-definition-admissible-blowup} and Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}). Thus the blowing up $X' \to X$ of $Y$ is a $U$-admissible blowing up. By general properties of strict transforms, the strict transform of $Z_1, Z_2$ with respect to $X' \to X$ is the same as the strict transform of $Z_1, Z_2$ with respect to $Z' \to Z$, see Divisors, Lemma \ref{divisors-lemma-strict-transform}. Thus (1) is proved. \medskip\noindent Proof of (2). By Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals} there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J}_i \subset \mathcal{O}_U$ such that $T_i = V(\mathcal{J}_i)$ (set theoretically). By Properties, Lemma \ref{properties-lemma-extend} there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I}_i \subset \mathcal{O}_X$ whose restriction to $U$ is $\mathcal{J}_i$. Apply the result of part (1) to the closed subschemes $Z_i = V(\mathcal{I}_i)$ to conclude.
Let $T' \subset V$ be the complement of the maximal open over which $f|_U$ is an isomorphism. Then $T', T$ are closed in $V$ and $T \cap T' = \emptyset$. Since $V$ is a spectral topological space, we can find constructible closed subsets $T_c, T'_c$ with $T \subset T_c$, $T' \subset T'_c$ such that $T_c \cap T'_c = \emptyset$ (choose a quasi-compact open $W$ of $V$ containing $T'$ not meeting $T$ and set $T_c = V \setminus W$, then choose a quasi-compact open $W'$ of $V$ containing $T_c$ not meeting $T'$ and set $T'_c = V \setminus W'$). By Lemma \ref{lemma-separate-disjoint-locally-closed-by-blowing-up} we may, after replacing $Y$ by a $V$-admissible blowing up, assume that $T_c$ and $T'_c$ have disjoint closures in $Y$. Set $Y_0 = Y \setminus \overline{T}'_c$, $V_0 = V \setminus T'_c$, $U_0 = U \times_V V_0$, and $X_0 = X \times_Y Y_0$. Since $U_0 \to V_0$ is an isomorphism, we can find a $V_0$-admissible blowing up $Y'_0 \to Y_0$ such that the strict transform $X'_0$ of $X_0$ maps isomorphically to $Y'_0$, see Lemma \ref{lemma-zariski-after-blowup}. By Divisors, Lemma \ref{divisors-lemma-extend-admissible-blowups} there exists a $V$-admissible blow up $Y' \to Y$ whose restriction to $Y_0$ is $Y'_0 \to Y_0$. If $f' : X' \to Y'$ denotes the strict transform of $f$, then we see what we want is true because $f'$ restricts to an isomorphism over $Y'_0$.
Throughout the proof all schemes will be separated of finite type over $S$. This in particular implies these schemes are quasi-compact and quasi-separated and the morphisms between them are quasi-compact and separated. See Schemes, Sections \ref{schemes-section-quasi-compact} and \ref{schemes-section-separation-axioms}. We will use that if $U \to W$ is an immersion of such schemes over $S$, then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subscheme of $W$ and $U \to Z$ is an open immersion, $U \subset Z$ is scheme theoretically dense, and $U \subset Z$ is dense topologically. See Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-immersion}. \medskip\noindent Let $X_{12} \subset X_1 \times_S X_2$ be the scheme theoretic image of $U \to X_1 \times_S X_2$. The projections $p_i : X_{12} \to X_i$ induce isomorphisms $p_i^{-1}(U) \to U$ by Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}. Choose a $U$-admissible blowup $X_i^i \to X_i$ such that the strict transform $X_{12}^i$ of $X_{12}$ is isomorphic to an open subscheme of $X_i^i$, see Lemma \ref{lemma-zariski-after-blowup}. Let $\mathcal{I}_i \subset \mathcal{O}_{X_i}$ be the corresponding finite type quasi-coherent sheaf of ideals. Recall that $X_{12}^i \to X_{12}$ is the blowup in $p_i^{-1}\mathcal{I}_i \mathcal{O}_{X_{12}}$, see Divisors, Lemma \ref{divisors-lemma-strict-transform}. Let $X_{12}'$ be the blowup of $X_{12}$ in $p_1^{-1}\mathcal{I}_1 p_2^{-1}\mathcal{I}_2 \mathcal{O}_{X_{12}}$, see Divisors, Lemma \ref{divisors-lemma-blowing-up-two-ideals} for what this entails. We obtain in particular a commutative diagram $$ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_{12}^2 \ar[d] \\ X_{12}^1 \ar[r] & X_{12} } $$ where all the morphisms are $U$-admissible blowing ups. Since $X_{12}^i \subset X_i^i$ is an open we may choose a $U$-admissible blowup $X_i' \to X_i^i$ restricting to $X_{12}' \to X_{12}^i$, see Divisors, Lemma \ref{divisors-lemma-extend-admissible-blowups}. Then $X_{12}' \subset X_i'$ is an open subscheme and the diagram $$ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_i' \ar[d] \\ X_{12}^i \ar[r] & X_i^i } $$ is commutative with vertical arrows blowing ups and horizontal arrows open immersions. Note that $X'_{12} \to X_1' \times_S X_2'$ is an immersion and proper (use that $X'_{12} \to X_{12}$ is proper and $X_{12} \to X_1 \times_S X_2$ is closed and $X_1' \times_S X_2' \to X_1 \times_S X_2$ is separated and apply Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). Thus $X'_{12} \to  X_1' \times_S X_2'$ is a closed immersion. It follows that if we define $X$ by glueing $X_1'$ and $X_2'$ along the common open subscheme $X_{12}'$, then $X \to S$ is of finite type and separated (Lemma \ref{lemma-check-separated}). As compositions of $U$-admissible blowups are $U$-admissible blowups (Divisors, Lemma \ref{divisors-lemma-composition-admissible-blowups}) the lemma is proved.
Denote $Z' \to Z$ the strict transform. The morphism $Z' \to X'$ induces a morphism $Z' \to X' \times_S Y$ which is a closed immersion (as $Z'$ is a closed subscheme of $X' \times_X Z$ by definition). Thus to finish the proof it suffices to show that the scheme theoretic image $Z''$ of $V \to Z'$ is $Z'$. Observe that $Z'' \subset Z'$ is a closed subscheme such that $V \to Z'$ factors through $Z''$. Since both $V \to X \times_S Y$ and $V \to X' \times_S Y$ are quasi-compact (for the latter this follows from Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence} and the fact that $X' \times_S Y \to X \times_S Y$ is separated as a base change of a proper morphism), by Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image} we see that $Z \cap (U \times_S Y) = Z'' \cap (U \times_S Y)$. Thus the inclusion morphism $Z'' \to Z'$ is an isomorphism away from the exceptional divisor $E$ of $Z' \to Z$. However, the structure sheaf of $Z'$ does not have any nonzero sections supported on $E$ (by definition of strict transforms) and we conclude that the surjection $\mathcal{O}_{Z'} \to \mathcal{O}_{Z''}$ must be an isomorphism.
Consider the scheme theoretic image $Z \subset Y \times_S U$ of the ``diagonal'' morphism $V \to Y \times_S U$. If we replace $Y$ by a $V$-admissible blowing up, then $Z$ is replaced by the strict transform with respect to this blowing up, see Lemma \ref{lemma-replaced-by-strict-transform}. Hence by Lemma \ref{lemma-zariski-after-blowup} we may assume $Z \to Y$ is an open immersion. If $V' \subset Y$ denotes the image, then we see that the induced morphism $V' \to U$ is proper because the projection $Y \times_S U \to U$ is proper and $V' \cong Z$ is a closed subscheme of $Y \times_S U$.
By assumption there exists a proper surjective morphism $Y \to X$ and a finite affine open covering $Y = \bigcup_{j = 1, \ldots, m} Y_j$ such that $\{Y_j \to X\}_{j = 1, \ldots, m}$ refines $\{X_i \to X\}_{i \in I}$. This means that for each $j$ there is an index $i_j \in I$ and a morphism $h_j : Y_j \to X_{i_j}$ over $X$. See Definition \ref{definition-h-covering} and Topologies, Definition \ref{topologies-definition-ph-covering}. The problem is that we don't know that $Y \to X$ is of finite presentation. By Limits, Lemma \ref{limits-lemma-proper-limit-of-proper-finite-presentation} we can write $$ Y = \lim Y_\lambda $$ as a directed limit of schemes $Y_\lambda$ proper and of finite presentation over $X$ such that the morphisms $Y \to Y_\lambda$ and the the transition morphisms are closed immersions. Observe that each $Y_\lambda \to X$ is surjective. By Limits, Lemma \ref{limits-lemma-descend-opens} we can find a $\lambda$ and quasi-compact opens $Y_{\lambda, j} \subset Y_\lambda$, $j = 1, \ldots, m$ covering $Y_\lambda$ and restricting to $Y_j$ in $Y$. Then $Y_j = \lim Y_{\lambda, j}$. After increasing $\lambda$ we may assume $Y_{\lambda, j}$ is affine for all $j$, see Limits, Lemma \ref{limits-lemma-limit-affine}. Finally, since $X_i \to X$ is locally of finite presentation we can use the functorial characterization of morphisms which are locally of finite presentation (Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation}) to find a $\lambda$ such that for each $j$ there is a morphism $h_{\lambda, j} : Y_{\lambda, j} \to X_{i_j}$ whose restriction to $Y_j$ is the morphism $h_j$ chosen above. Thus $\{Y_{\lambda, j} \to X\}$ refines $\{X_i \to X\}$ and the proof is complete.
This is true because in an fppf covering the morphisms are required to be locally of finite presentation and because fppf coverings are ph covering, see More on Morphisms, Lemma \ref{more-morphisms-lemma-fppf-ph}. The second statement follows from the first and Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf}.
Reasoning as in the proof of Topologies, Lemma \ref{topologies-lemma-verify-site-etale} it suffices to show that the collection of standard h coverings satisfies properties (1), (2) and (3) of Sites, Definition \ref{sites-definition-site}. This is clear since for example, given a standard h covering $\{T_i \to T\}_{i\in I}$ and for each $i$ a standard h covering $\{T_{ij} \to T_i\}_{j \in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a h covering (Lemma \ref{lemma-h}), $\bigcup_{i\in I} J_i$ is finite and each $T_{ij}$ is affine. Thus $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a standard h covering.
For $\Sch_h$ it is true by construction, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \Ob(\Sch_h)$. The fibre product $V \times_U W$ in $\Sch_h$ is a fibre product in $\Sch$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\Sch/S)_h$. This proves the result for $(\Sch/S)_h$. If $U, V, W$ are affine, so is $V \times_U W$ and hence the result for $(\textit{Aff}/S)_h$.
The notion of a special cocontinuous functor is introduced in Sites, Definition \ref{sites-definition-special-cocontinuous-functor}. Thus we have to verify assumptions (1) -- (5) of Sites, Lemma \ref{sites-lemma-equivalence}. Denote the inclusion functor $u : (\textit{Aff}/S)_h \to (\Sch/S)_h$. Being cocontinuous follows because any h covering of $T/S$, $T$ affine, can be refined by a standard h covering for example by Lemma \ref{lemma-approximate-h-cover}. Hence (1) holds. We see $u$ is continuous simply because a standard h covering is a h covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering (which is a h covering).
We will show that if (1) and (2) hold, then $\mathcal{F}$ is sheaf. Let $\{T_i \to T\}$ be a covering in $(\Sch/S)_h$. We will verify the sheaf condition for this covering. Let $s_i \in \mathcal{F}(T_i)$ be sections which restrict to the same section over $T_i \times_T T_{i'}$. We will show that there exists a unique section $s \in \mathcal{F}(T)$ restricting to $s_i$ over $T_i$. Let $T = \bigcup U_j$ be an affine open covering. By property (1) it suffices to produce sections $s_j \in \mathcal{F}(U_j)$ which agree on $U_j \cap U_{j'}$ in order to produce $s$. Consider the coverings $\{T_i \times_T U_j \to U_j\}$. Then $s_{ji} = s_i|_{T_i \times_T U_j}$ are sections agreeing over $(T_i \times_T U_j) \times_{U_j} (T_{i'} \times_T U_j)$. Choose a proper surjective morphism $V_j \to U_j$ of finite presentation and a finite affine open covering $V_j = \bigcup V_{jk}$ such that $\{V_{jk} \to U_j\}$ refines $\{T_i \times_T U_j \to U_j\}$. See Lemma \ref{lemma-approximate-h-cover}. If $s_{jk} \in \mathcal{F}(V_{jk})$ denotes the pullback of $s_{ji}$ to $V_{jk}$ by the implied morphisms, then we find that $s_{jk}$ glue to a section $s'_j \in \mathcal{F}(V_j)$. Using the agreement on overlaps once more, we find that $s'_j$ is in the equalizer of the two maps $\mathcal{F}(V_j) \to \mathcal{F}(V_j \times_{U_j} V_j)$. Hence by (2) we find that $s'_j$ comes from a unique section $s_j \in \mathcal{F}(U_j)$. We omit the verification that these sections $s_j$ have all the desired properties. \medskip\noindent Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $V \to U$ is a morphism of $(\Sch/S)_h$ which is proper, surjective, and of finite presentation. Choose an affine open covering $U = \bigcup U_i$ and set $V_i = V \times_U U_i$. Then we see that $\mathcal{F}(U) \to \mathcal{F}(V)$ is injective because we know $\mathcal{F}(U_i) \to \mathcal{F}(V_i)$ is injective by (2') and we know $\mathcal{F}(U) \to \prod \mathcal{F}(U_i)$ is injective by (1). Finally, suppose that we are given an $t \in \mathcal{F}(V)$ in the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times_U V)$. Then $t|_{V_i}$ is in the equalizer of the two maps $\mathcal{F}(V_i) \to \mathcal{F}(V_i \times_{U_i} V_i)$ for all $i$. Hence we obtain a unique section $s_i \in \mathcal{F}(U_i)$ mapping to $t|_{V_i}$ for all $i$ by (2'). We omit the verification that $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$; this uses the uniqueness property just shown. By the sheaf property for the covering $U = \bigcup U_i$ we obtain a section $s \in \mathcal{F}(U)$. We omit the proof that $s$ maps to $t$ in $\mathcal{F}(V)$.
The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-preserve-equalizers} apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/S$ we have $\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)$ as desired. Thus we may apply Sites, Lemmas \ref{sites-lemma-have-functor-other-way} and \ref{sites-lemma-have-functor-other-way-morphism} to get the formula for $f_{big, *}$.
By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} the category of schemes of finite presentation over $T$ is the colimit over $I$ of the categories of finite presentation over $T_i$. By Limits, Lemma \ref{limits-lemma-descend-affine-finite-presentation} the same is true for category of schemes which are affine and of finite presentation over $T$. To finish the proof of the lemma it suffices to show that if $\{V_{j, i} \to T_i\}_{j = 1, \ldots, m}$ is a finite family of finitely presented morphisms with $V_{j, i}$ affine, and the base change family $\{T \times_{T_i} V_{j, i} \to T\}$ is an h covering, then for some $i' \geq i$ the family $\{T_{i'} \times_{T_i} V_{j, i} \to T_{i'}\}$ is an h covering. To see this we use Lemma \ref{lemma-approximate-h-cover} to choose a finitely presented, proper, surjective morphism $Y \to T$ and a finite affine open covering $Y = \bigcup_{k = 1, \ldots, n} Y_k$ such that $\{Y_k \to T\}_{k = 1, \ldots, n}$ refines $\{T \times_{T_i} V_{j, i} \to T\}$. Using the arguments above and Limits, Lemmas \ref{limits-lemma-eventually-proper}, \ref{limits-lemma-descend-surjective}, and \ref{limits-lemma-descend-opens} we can find an $i' \geq i$ and a finitely presented, surjective, proper morphism $Y_{i'} \to T_{i'}$ and an affine open covering $Y_{i'} = \bigcup_{k = 1, \ldots, n} Y_{i', k}$ such that moreover $\{Y_{i', k} \to Y_{i'}\}$ refines $\{T_{i'} \times_{T_i} V_{j, i} \to T_{i'}\}$. It follows that this last mentioned family is a h covering and the proof is complete.
Since $Z \amalg X' \to X$ is a surjective proper morphism we see that $\{Z \amalg X' \to X\}$ is a ph covering (Topologies, Lemma \ref{topologies-lemma-surjective-proper-ph}). We have $$ (Z \amalg X') \times_X (Z \amalg X') = Z \amalg E \amalg E \amalg X' \times_X X' $$ Since $\mathcal{F}$ is a Zariski sheaf we see that $\mathcal{F}$ sends disjoint unions to products. Thus the sheaf condition for the covering $\{Z \amalg X' \to X\}$ says that $\mathcal{F}(X) \to \mathcal{F}(Z) \times \mathcal{F}(X')$ is injective with image the set of pairs $(t, s')$ such that (a) $t|_E = s'|_E$ and (b) $s'$ is in the equalizer of the two maps $\mathcal{F}(X') \to \mathcal{F}(X' \times_X X')$. Next, observe that the obvious morphism $$ E \times_Z E \amalg X' \longrightarrow X' \times_X X' $$ is a surjective proper morphism as $b$ induces an isomorphism $X' \setminus E \to X \setminus Z$. We conclude that $\mathcal{F}(X' \times_X X') \to \mathcal{F}(E \times_Z E) \times \mathcal{F}(X')$ is injective. It follows that (a) $\Rightarrow$ (b) which means that the lemma is true.
The morphism $Y \times_X X' \to Y$ is proper and of finite presentation by Morphisms, Lemmas \ref{morphisms-lemma-base-change-proper} and \ref{morphisms-lemma-base-change-finite-presentation}. The morphism $Y \times_X Z \to Y$ is a closed immersion (Morphisms, Lemma \ref{morphisms-lemma-base-change-closed-immersion}) of finite presentation. The inverse image of $Y \times_X Z$ in $Y \times_X X'$ is equal to the inverse image of $E$ in $Y \times_X X'$ and hence is locally principal (Divisors, Lemma \ref{divisors-lemma-pullback-locally-principal}). Let $X'' \subset X'$, resp.\ $Y'' \subset Y \times_X X'$ be the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{X'}$, resp.\ $\mathcal{O}_{Y \times_Y X'}$ supported on $E$, resp.\ $Y \times_X E$. Clearly, $Y'' \subset Y \times_X X''$ is the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{Y \times_Y X''}$ supported on $Y \times_X (E \cap X'')$. Thus $Y''$ is the strict transform of $Y$ relative to the blowing up $X'' \to X$, see Divisors, Definition \ref{divisors-definition-strict-transform}. Thus by Divisors, Lemma \ref{divisors-lemma-strict-transform} we see that $Y''$ is the blow up of $Y \times_X Z$ on $Y$.
Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent sheaf of ideals corresponding to $X''$. By Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} we can write $\mathcal{I}$ as the filtered colimit $\mathcal{I} = \colim \mathcal{I}_i$ of its quasi-coherent submodules of finite type. Since these modules correspond $1$-to-$1$ to the closed subschemes $X'_i$ the proof is complete.
We may write $X = \lim X_i$ as a directed limit of an inverse system of Noetherian schemes with affine transition morphisms, see Limits, Proposition \ref{limits-proposition-approximate}. We can find an index $i$ and a closed immersion $Z_i \to X_i$ whose base change to $X$ is the closed immersion $Z \to X$. See Limits, Lemmas \ref{limits-lemma-descend-finite-presentation} and \ref{limits-lemma-descend-closed-immersion-finite-presentation}. Let $b_i : X'_i \to X_i$ be the blowing up with center $Z_i$. This produces a blow up square $$ \xymatrix{ E_i \ar[r] \ar[d] & X'_i \ar[d]^{b_i} \\ Z_i \ar[r] & X_i } $$ where all the morphisms are finite type morphisms of Noetherian schemes and hence of finite presentation. Thus this is an almost blow up square. By Lemma \ref{lemma-base-change-almost-blow-up} the base change of this diagram to $X$ produces the desired almost blow up square.
Denote $X'' \to X$ the blowing up of $Z$ in $X$. We view $X''$ as a closed subscheme of both $X'_1$ and $X'_2$. Write $X'' = \lim X'_{1, i}$ as in Lemma \ref{lemma-blow-up-limit-almost-blow-up}. By Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} there exists an $i$ and a morphism $h : X'_{1, i} \to X'_2$ agreeing with the inclusions $X'' \subset X'_{1, i}$ and $X'' \subset X'_2$. By Limits, Lemma \ref{limits-lemma-eventually-closed-immersion} the restriction of $h$ to $X'_{1, i'}$ is a closed immersion for some $i' \geq i$. This finishes the proof.
If $Y$ is a Noetherian scheme, then this lemma follows immediately from Lemma \ref{lemma-flat-after-blowing-up} because in this case blow up squares are almost blow up squares (we also use that strict transforms are blow ups). The general case is reduced to the Noetherian case by absolute Noetherian approximation. \medskip\noindent We may write $Y = \lim Y_i$ as a directed limit of an inverse system of Noetherian schemes with affine transition morphisms, see Limits, Proposition \ref{limits-proposition-approximate}. We can find an index $i$ and a morphism $X_i \to Y_i$ of finite presentation whose base change to $Y$ is $X \to Y$. See Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}. After increasing $i$ we may assume $V$ is the inverse image of an open subscheme $V_i \subset Y_i$, see Limits, Lemma \ref{limits-lemma-descend-opens}. Finally, after increasing $i$ we may assume that $X_{i, V_i} \to V_i$ is flat, see Limits, Lemma \ref{limits-lemma-descend-flat-finite-presentation}. By the Noetherian case, we may construct a diagram as in the lemma for $X_i \to Y_i \supset V_i$. The base change of this diagram by $Y \to Y_i$ provides the solution. Use that base change preserves properties of morphisms, see Morphisms, Lemmas \ref{morphisms-lemma-base-change-proper}, \ref{morphisms-lemma-base-change-finite-presentation}, \ref{morphisms-lemma-base-change-closed-immersion}, and \ref{morphisms-lemma-base-change-flat} and that base change of an almost blow up square is an almost blow up square, see Lemma \ref{lemma-base-change-almost-blow-up}.