hcju/stacksqa huggingface dataset:
Let $f : X \to Y$ be a flat monomorphism of algebraic spaces. To prove $f$ is representable, we have to show $X \times_Y V$ is a scheme for every scheme $V$ mapping to $Y$. Since being a scheme is local (Properties of Spaces,  Lemma \ref{spaces-properties-lemma-subscheme}), we may assume $V$ is affine. Thus we may assume $Y = \Spec(B)$ is an affine scheme. Next, we can assume that $X$ is quasi-compact by replacing $X$ by a quasi-compact open. The space $X$ is separated as $X \to X \times_{\Spec(B)} X$ is an isomorphism. Applying Limits of Spaces, Lemma \ref{spaces-limits-lemma-enough-local} we reduce to the case where $B$ is local, $X \to \Spec(B)$ is a flat monomorphism, and there exists a point $x \in X$ mapping to the closed point of $\Spec(B)$. Then $X \to \Spec(B)$ is surjective as generalizations lift along flat morphisms of separated algebraic spaces, see Decent Spaces, Lemma \ref{decent-spaces-lemma-generalizations-lift-flat}. Hence we see that $\{X \to \Spec(B)\}$ is an fpqc cover. Then $X \to \Spec(B)$ is a morphism which becomes an isomorphism after base change by $X \to \Spec(B)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-isomorphism}.
The question is \'etale local on $Y$, hence we may assume $Y = \Spec(A)$ is affine. Then $X$ is quasi-compact and we may choose an affine scheme $U = \Spec(B)$ and a surjective \'etale morphism $U \to X$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Note that $U \times_X U = \Spec(B \otimes_A B)$. Hence the category of quasi-coherent $\mathcal{O}_X$-modules is equivalent to the category $DD_{B/A}$ of descent data on modules for $A \to B$. See Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent}, Descent, Definition \ref{descent-definition-descent-datum-modules}, and Descent, Subsection \ref{descent-subsection-descent-modules-morphisms}. On the other hand, $$ A \to B $$ is a universally injective ring map. Namely, given an $A$-module $M$ we see that $A \oplus M \to B \otimes_A (A \oplus M)$ is injective by the assumption of the lemma. Hence $DD_{B/A}$ is equivalent to the category of $A$-modules by Descent, Theorem \ref{descent-theorem-descent}. Thus pullback along $f : X \to \Spec(A)$ determines an equivalence of categories of quasi-coherent modules. In particular $f^*$ is exact on quasi-coherent modules and we see that $f$ is flat (small detail omitted). Moreover, it is clear that $f$ is surjective (for example because $\Spec(B) \to \Spec(A)$ is surjective). Hence we see that $\{X \to \Spec(A)\}$ is an fpqc cover. Then $X \to \Spec(A)$ is a morphism which becomes an isomorphism after base change by $X \to \Spec(A)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-isomorphism}.
Let $T \subset X$ be a closed subspace such that $i$ defines a closed immersion into $X \setminus T$. Let $\mathcal{I}$ be the quasi-coherent sheaf of ideals on $X \setminus T$ defining $Z$. Then the lemma just states that $\mathcal{I}|_{U \setminus \varphi^{-1}(T)}$ is the sheaf of ideals of the immersion $Z_U \to U \setminus \varphi^{-1}(T)$. This is clear from the construction of $\mathcal{I}$ in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}.
First find open subspaces $U' \subset X'$ and $U \subset X$ such that $g(U) \subset U'$ and such that $i(Z) \subset U$ and $i(Z') \subset U'$ are closed (proof existence omitted). Replacing $X$ by $U$ and $X'$ by $U'$ we may assume that $i$ and $i'$ are closed immersions. Let $\mathcal{I}' \subset \mathcal{O}_{X'}$ and $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaves of ideals associated to $i'$ and $i$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}. Consider the composition $$ g^{-1}\mathcal{I}' \to g^{-1}\mathcal{O}_{X'} \xrightarrow{g^\sharp} \mathcal{O}_X \to \mathcal{O}_X/\mathcal{I} = i_*\mathcal{O}_Z $$ Since $g(i(Z)) \subset Z'$ we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}). Thus we obtain a commutative diagram $$ \xymatrix{ 0 \ar[r] & \mathcal{I} \ar[r] & \mathcal{O}_X \ar[r] & i_*\mathcal{O}_Z \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{I}' \ar[r] \ar[u] & g^{-1}\mathcal{O}_{X'} \ar[r] \ar[u] & g^{-1}i'_*\mathcal{O}_{Z'} \ar[r] \ar[u] & 0 } $$ The lower row is exact since $g^{-1}$ is an exact functor. By exactness we also see that $(g^{-1}\mathcal{I}')^2 = g^{-1}((\mathcal{I}')^2)$. Hence the diagram induces a map $g^{-1}(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{I}/\mathcal{I}^2$. Pulling back (using $i^{-1}$ for example) to $Z$ we obtain $i^{-1}g^{-1}(\mathcal{I}'/(\mathcal{I}')^2) \to \mathcal{C}_{Z/X}$. Since $i^{-1}g^{-1} = f^{-1}(i')^{-1}$ this gives a map $f^{-1}\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$, which induces the desired map.
Choose a commutative diagram $$ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ U' \ar[r] & X' } $$ where $U$, $U'$ are schemes and the horizontal arrows are surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. Then using Lemmas \ref{lemma-etale-conormal} and \ref{lemma-conormal-functorial-more} we see that the question reduces to the case of a morphism of schemes. In the schemes case this is Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial-flat}.
Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. Via Lemmas \ref{lemma-etale-conormal} and \ref{lemma-conormal-functorial-more} the exactness of the sequence translates immediately into the exactness of the corresponding sequence for the immersions of schemes $Z \times_X U \to Y \times_X U \to U$. Hence the lemma follows from Morphisms, Lemma \ref{morphisms-lemma-transitivity-conormal}.
Let $T \subset X$ be a closed subspace such that $i$ defines a closed immersion into $X \setminus T$. Let $\mathcal{I}$ be the quasi-coherent sheaf of ideals on $X \setminus T$ defining $Z$. Then the lemma follows from the fact that $\mathcal{I}|_{U \setminus \varphi^{-1}(T)}$ is the sheaf of ideals of the immersion $Z_U \to U \setminus \varphi^{-1}(T)$. This is clear from the construction of $\mathcal{I}$ in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}.
First find open subspaces $U' \subset X'$ and $U \subset X$ such that $g(U) \subset U'$ and such that $i(Z) \subset U$ and $i(Z') \subset U'$ are closed (proof existence omitted). Replacing $X$ by $U$ and $X'$ by $U'$ we may assume that $i$ and $i'$ are closed immersions. Let $\mathcal{I}' \subset \mathcal{O}_{X'}$ and $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaves of ideals associated to $i'$ and $i$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}. Consider the composition $$ g^{-1}\mathcal{I}' \to g^{-1}\mathcal{O}_{X'} \xrightarrow{g^\sharp} \mathcal{O}_X \to \mathcal{O}_X/\mathcal{I} = i_*\mathcal{O}_Z $$ Since $g(i(Z)) \subset Z'$ we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}). Thus we obtain a commutative diagram $$ \xymatrix{ 0 \ar[r] & \mathcal{I} \ar[r] & \mathcal{O}_X \ar[r] & i_*\mathcal{O}_Z \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{I}' \ar[r] \ar[u] & g^{-1}\mathcal{O}_{X'} \ar[r] \ar[u] & g^{-1}i'_*\mathcal{O}_{Z'} \ar[r] \ar[u] & 0 } $$ The lower row is exact since $g^{-1}$ is an exact functor. By exactness we also see that $(g^{-1}\mathcal{I}')^n = g^{-1}((\mathcal{I}')^n)$ for all $n \geq 1$. Hence the diagram induces a map $g^{-1}((\mathcal{I}')^n/(\mathcal{I}')^{n + 1}) \to \mathcal{I}^n/\mathcal{I}^{n + 1}$. Pulling back (using $i^{-1}$ for example) to $Z$ we obtain $i^{-1}g^{-1}((\mathcal{I}')^n/(\mathcal{I}')^{n + 1}) \to \mathcal{C}_{Z/X, n}$. Since $i^{-1}g^{-1} = f^{-1}(i')^{-1}$ this gives maps $f^{-1}\mathcal{C}_{Z'/X', n} \to \mathcal{C}_{Z/X, n}$, which induce the desired map.
Let $h : U \to X$ be an \'etale morphism. In this case the natural map $h^*\Omega_{X/Y} \to \Omega_{U/Y}$ is an isomorphism, see More on Morphisms, Lemma \ref{more-morphisms-lemma-sheaf-differentials-etale-localization}. This means that there is a natural $\mathcal{O}_{Y_\etale}$-derivation $$ \text{d}^a : \mathcal{O}_{X_\etale} \longrightarrow (\Omega_{X/Y})^a $$ since we have just seen that the value of $(\Omega_{X/Y})^a$ on any object $U$ of $X_\etale$ is canonically identified with $\Gamma(U, \Omega_{U/Y})$. By the universal property of $\text{d}_{X/Y} : \mathcal{O}_{X_\etale} \to \Omega_{X_\etale/Y_\etale}$ there is a unique $\mathcal{O}_{X_\etale}$-linear map $c : \Omega_{X_\etale/Y_\etale} \to (\Omega_{X/Y})^a$ such that $\text{d}^a = c \circ \text{d}_{X/Y}$. \medskip\noindent Conversely, suppose that $\mathcal{F}$ is an $\mathcal{O}_{X_\etale}$-module and $D : \mathcal{O}_{X_\etale} \to \mathcal{F}$ is a $\mathcal{O}_{Y_\etale}$-derivation. Then we can simply restrict $D$ to the small Zariski site $X_{Zar}$ of $X$. Since sheaves on $X_{Zar}$ agree with sheaves on $X$, see Descent, Remark \ref{descent-remark-Zariski-site-space}, we see that $D|_{X_{Zar}} : \mathcal{O}_X \to \mathcal{F}|_{X_{Zar}}$ is just a ``usual'' $Y$-derivation. Hence we obtain a map $\psi : \Omega_{X/Y} \longrightarrow \mathcal{F}|_{X_{Zar}}$ such that $D|_{X_{Zar}} = \psi \circ \text{d}$. In particular, if we apply this with $\mathcal{F} = \Omega_{X_\etale/Y_\etale}$ we obtain a map $$ c' : \Omega_{X/Y} \longrightarrow \Omega_{X_\etale/Y_\etale}|_{X_{Zar}} $$ Consider the morphism of ringed sites $\text{id}_{small, \etale, Zar} : X_\etale \to X_{Zar}$ discussed in Descent, Remark \ref{descent-remark-change-topologies-ringed} and Lemma \ref{descent-lemma-compare-sites}. Since the restriction functor $\mathcal{F} \mapsto \mathcal{F}|_{X_{Zar}}$ is equal to $\text{id}_{small, \etale, Zar, *}$, since $\text{id}_{small, \etale, Zar}^*$ is left adjoint to $\text{id}_{small, \etale, Zar, *}$ and since $(\Omega_{X/Y})^a = \text{id}_{small, \etale, Zar}^*\Omega_{X/Y}$ we see that $c'$ is adjoint to a map $$ c'' : (\Omega_{X/Y})^a \longrightarrow \Omega_{X_\etale/Y_\etale}. $$ We claim that $c''$ and $c'$ are mutually inverse. This claim finishes the proof of the lemma. To see this it is enough to show that $c''(\text{d}(f)) = \text{d}_{X/Y}(f)$ and $c(\text{d}_{X/Y}(f)) = \text{d}(f)$ if $f$ is a local section of $\mathcal{O}_X$ over an open of $X$. We omit the verification.
By Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-morphism-topoi} and Equation (\ref{spaces-properties-equation-restrict}) we may think of the restriction of a sheaf on $X_\etale$ to $U_\etale$ as the pullback by $a_{small}$. Similarly for $b$. By Modules on Sites, Lemma \ref{sites-modules-lemma-localize-differentials} we have $$ \Omega_{X/Y}|_{U_\etale} = \Omega_{\mathcal{O}_{U_\etale}/ a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_\etale}} $$ Since $a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_\etale} = \psi_{small}^{-1}b_{small}^{-1}\mathcal{O}_{Y_\etale} = \psi_{small}^{-1}\mathcal{O}_{V_\etale}$ we see that the lemma holds.
Choose a diagram as in Lemma \ref{lemma-localize-differentials} with $a$ and $b$ surjective and $U$ and $V$ schemes. Then we see that $\Omega_{X/Y}|_U = \Omega_{U/V}$ which is quasi-coherent (for example by Morphisms, Lemma \ref{morphisms-lemma-differentials-diagonal}). Hence we conclude that $\Omega_{X/Y}$ is quasi-coherent by Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent}.
This is the algebraic spaces version of Morphisms, Lemma \ref{morphisms-lemma-differentials-relative-immersion} and will be a consequence of that lemma by \'etale localization, see Lemmas \ref{lemma-localize-differentials} and \ref{lemma-etale-conormal}. However, we should make sure we can define the first arrow globally. Hence we explain the meaning of ``induced by $\text{d}_{X/B}$'' here. Namely, we may assume that $i$ is a closed immersion after replacing $X$ by an open subspace. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals corresponding to $Z \subset X$. Then $\text{d}_{X/S} : \mathcal{I} \to \Omega_{X/S}$ maps the subsheaf $\mathcal{I}^2 \subset \mathcal{I}$ to $\mathcal{I}\Omega_{X/S}$. Hence it induces a map $\mathcal{I}/\mathcal{I}^2 \to \Omega_{X/S}/\mathcal{I}\Omega_{X/S}$ which is $\mathcal{O}_X/\mathcal{I}$-linear. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence} this corresponds to a map $\mathcal{C}_{Z/X} \to i^*\Omega_{X/S}$ as desired.
Clarification: we claim that if $g : X \to Z$ is a left inverse of $i$ over $B$, then $i^*c_g$ is a right inverse of the map $i^*\Omega_{X/B} \to \Omega_{Z/B}$. Having said this, the result follows from the corresponding result for morphisms of schemes by \'etale localization, see Lemmas \ref{lemma-localize-differentials} and \ref{lemma-etale-conormal}.
This is just the statement that given an object $V \in Y_{spaces, \etale}$ we have $V$ is a(n affine) scheme if and only if $V \times_Y X$ is a(n affine) scheme. Since $V \times_Y X \to V$ is integral, universally injective, and surjective (as a base change of $X \to Y$) this follows from Limits of Spaces, Lemma \ref{spaces-limits-lemma-integral-universally-bijective-scheme} and Proposition \ref{spaces-limits-proposition-affine}.
To find $f'$, by Properties of Spaces, Theorem \ref{spaces-properties-theorem-fully-faithful}, all we have to do is show that the morphism of ringed topoi $$ (f_{spaces, \etale}, \alpha) : (\Sh(X_{spaces, \etale}), \mathcal{O}_{X'}) \longrightarrow (\Sh(Y_{spaces, \etale}), \mathcal{O}_{Y'}) $$ is a morphism of locally ringed topoi. This follows directly from the definition of morphisms of locally ringed topoi (Modules on Sites, Definition \ref{sites-modules-definition-morphism-locally-ringed-topoi}), the fact that $(f, f^\sharp)$ is a morphism of locally ringed topoi (Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-locally-ringed}), that $\alpha$ fits into the given commutative diagram, and the fact that the kernels of $i_X^\sharp$ and $i_Y^\sharp$ are locally nilpotent. Finally, the fact that $f' \circ i_X = i_Y \circ f$ follows from the commutativity of the diagram and another application of Properties of Spaces, Theorem \ref{spaces-properties-theorem-fully-faithful}. We omit the verification that $f'$ is a morphism over $B$.
Let $U' \to X'$ be the object of $X'_{spaces, \etale}$ corresponding to the object $U \to X$ of $X_{spaces, \etale}$ via (\ref{equation-equivalence-etale-spaces}). The morphism $U' \to X'$ is \'etale and universally injective, hence an open immersion, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open}.
Let $U' \to X'$ be the \'etale morphism of algebraic spaces such that $U = X \times_{X'} U'$, see Theorem \ref{theorem-topological-invariance}. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-affine} we see that $U'$ is an affine scheme. Hence $\Gamma(U, \mathcal{O}_{X'}) = \Gamma(U', \mathcal{O}_{U'}) \to \Gamma(U, \mathcal{O}_U)$ is surjective as $U \to U'$ is a closed immersion of affine schemes. Below we give a direct proof for finite order thickenings which is the case most used in practice.
The statement means the following: Let $S$ be a scheme and let $X \to X'$ be a morphism of algebraic spaces over $S$. Let $\{g_i : X'_i \to X'\}$ be an fpqc covering of algebraic spaces such that the base change $X_i \to X'_i$ is a thickening for all $i$. Then $X \to X'$ is a thickening. Since the morphisms $g_i$ are jointly surjective we conclude that $X \to X'$ is surjective. By Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-closed-immersion} we conclude that $X \to X'$ is a closed immersion. Thus $X \to X'$ is a thickening. We omit the proof in the case of first order thickenings.
Observe that $Y \to Y'$ and $X \to X'$ are integral and universal homeomorphisms. This immediately implies parts (2), (3), (4), (7), and (8). Part (1) follows from Limits of Spaces, Proposition \ref{spaces-limits-proposition-affine} which tells us that there is a 1-to-1 correspondence between affine schemes \'etale over $X$ and $X'$ and between affine schemes \'etale over $Y$ and $Y'$. Part (5) follows from (1) and (4) by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-universally-closed}. Finally, note that $$ X \times_Y X = X \times_{Y'} X \to X \times_{Y'} X' \to X' \times_{Y'} X' $$ is a thickening (the two arrows are thickenings by Lemma \ref{lemma-base-change-thickening}). Hence applying (3) and (4) to the morphism $(X \subset X') \to (X \times_Y X \to X' \times_{Y'} X')$ we obtain (6). Finally, part (9) follows from the fact that an algebraic space thickening of a scheme is again a scheme, see Lemma \ref{lemma-thickening-scheme}.
Recall that $X_\etale = X'_\etale$, see Lemma \ref{lemma-thickening-equivalence} and more generally the discussion in Section \ref{section-thickenings}. The sequence of the lemma is the long exact cohomology sequence associated to the short exact sequence of sheaves of abelian groups $$ 0 \to \mathcal{I} \to \mathcal{O}_{X'}^* \to \mathcal{O}_X^* \to 0 $$ on $X_\etale$ where the first map sends a local section $f$ of $\mathcal{I}$ to the invertible section $1 + f$ of $\mathcal{O}_{X'}$. We also use the identification of the Picard group of a ringed site with the first cohomology group of the sheaf of invertible functions, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-h1-invertible}.
Let $U \subset X$ be the open subspace used in the construction of $Z'$, i.e., an open such that $Z$ is identified with a closed subspace of $U$ cut out by the quasi-coherent sheaf of ideals $\mathcal{I}$. Since $|T| = |T'|$ we see that $|b|(|T'|) \subset |U|$. Hence we can think of $b$ as a morphism into $U$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-factor-through-open-subspace}. Let $\mathcal{J} \subset \mathcal{O}_{T'}$ be the square zero quasi-coherent sheaf of ideals cutting out $T$. By the commutativity of the diagram we have $b|_T = i \circ a$ where $i : Z \to U$ is the closed immersion. We conclude that $b^\sharp(b^{-1}\mathcal{I}) \subset \mathcal{J}$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}. As $T'$ is a first order thickening of $T$ we see that $\mathcal{J}^2 = 0$ hence $b^\sharp(b^{-1}(\mathcal{I}^2)) = 0$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals} this implies that $b$ factors through $Z'$. Letting $a' : T' \to Z'$ be this factorization we win.
Consider a solid commutative diagram $$ \xymatrix{ F \ar[d]_a & T \ar[d]^i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] } $$ as in Definition \ref{definition-formally-smooth-etale-unramified}. Then $F \times_G T'$ is a scheme smooth (resp.\ \'etale, resp.\ unramified) over $T'$. Hence by More on Morphisms, Lemma \ref{more-morphisms-lemma-smooth-formally-smooth} (resp.\ Lemma \ref{more-morphisms-lemma-etale-formally-etale}, resp.\ Lemma \ref{more-morphisms-lemma-unramified-formally-unramified}) we can fill in (resp.\ uniquely fill in, resp.\ fill in at most one way) the dotted arrow in the diagram $$ \xymatrix{ F \times_G T' \ar[d] & T \ar[d]^i \ar[l] \\ T' & T' \ar[l] \ar@{-->}[lu] } $$ an hence we also obtain the corresponding assertion in the first diagram.
Assume $b$ is formally smooth (resp.\ formally \'etale, resp.\ formally unramified). Since an \'etale morphism is both smooth and unramified we see that $a$ is representable and smooth (resp.\ \'etale, resp. unramified). Hence parts (1), (2) and (3) follow from a combination of Lemma \ref{lemma-representable-property-formally-property} and Lemma \ref{lemma-composition-formally-smooth-etale-unramified}. \medskip\noindent Assume that $b \circ a$ is formally smooth. Consider a diagram as in the statement of the lemma. Let $W = F \times_G T$. By assumption $W$ is a scheme surjective \'etale over $T$. By \'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence} there exists a scheme $W'$ \'etale over $T'$ such that $W = T \times_{T'} W'$. Choose an affine open subscheme $U' \subset W'$ such that $t$ is in the image of $U' \to T'$. Because $b \circ a$ is formally smooth we see that the exist morphisms $U' \to F$ such that $$ \xymatrix{ F \ar[d]_{b \circ a} & W \ar[l] & T \times_{T'} U' \ar[d] \ar[l]\\ H & T' \ar[l] & U' \ar[llu] \ar[l] } $$ commutes. Taking the composition $U' \to F \to G$ gives a map as in part (5) of the lemma. \medskip\noindent Assume that $f, g : T' \to G$ are two dotted arrows fitting into the diagram of the lemma. Let $W = F \times_G T$. By assumption $W$ is a scheme surjective \'etale over $T$. By \'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence} there exists a scheme $W'$ \'etale over $T'$ such that $W = T \times_{T'} W'$. Since $a$ is formally \'etale the compositions $$ W' \to T' \xrightarrow{f} G \quad\text{and}\quad W' \to T' \xrightarrow{g} G $$ lift to morphisms $f', g' : W' \to F$ (lift on affine opens and glue by uniqueness). Now if $b \circ a : F \to H$ is formally unramified, then $f' = g'$ and hence $f = g$ as $W' \to T'$ is an \'etale covering. This proves part (6) of the lemma. \medskip\noindent Assume that $b \circ a$ is formally \'etale. Then by part (4) we can \'etale locally on $T'$ find a dotted arrow fitting into the diagram and by part (5) this dotted arrow is unique. Hence we may glue the local solutions to get assertion (6). Some details omitted.
Assume $f$ is formally unramified. By Lemma \ref{lemma-representable-property-formally-property} the morphisms $U \to X$ and $V \to Y$ are formally unramified. Thus by Lemma \ref{lemma-composition-formally-smooth-etale-unramified} the composition $U \to Y$ is formally unramified. Then it follows from Lemma \ref{lemma-formally-permanence} that $U \to V$ is formally unramified. Thus (1) implies (2). And (2) implies (3) trivially \medskip\noindent Assume given a diagram as in (3). By Lemma \ref{lemma-representable-property-formally-property} the morphism $V \to Y$ is formally unramified. Thus by Lemma \ref{lemma-composition-formally-smooth-etale-unramified} the composition $U \to Y$ is formally unramified. Then it follows from Lemma \ref{lemma-etale-on-top} that $X \to Y$ is formally unramified, i.e., (1) holds.
Choose a diagram $$ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \'etale and surjective. Then we see \begin{align*} f\text{ unramified} & \Leftrightarrow \psi\text{ unramified} \\ & \Leftrightarrow \psi\text{ locally finite type and }\Omega_{U/V} = 0 \\ & \Leftrightarrow f\text{ locally finite type and }\Omega_{X/Y} = 0 \\ & \Leftrightarrow f\text{ locally finite type and formally unramified} \end{align*} Here we have used Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero} and Lemma \ref{lemma-characterize-formally-unramified}.
We have seen in Lemma \ref{lemma-unramified-formally-unramified} that being formally unramified and locally of finite type is the same thing as being unramified. Hence (4) is equivalent to (2). A monomorphism is certainly formally unramified hence (3) implies (4). It is clear that (1) implies (3). Finally, if (2) holds, then $\Delta : X \to X \times_Y X$ is both an open immersion (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-unramified-morphism}) and surjective (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-injective}) hence an isomorphism, i.e., $f$ is a monomorphism. In this way we see that (2) implies (1). Finally, we see that $f$ is representable, separated, and locally quasi-finite by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite} and \ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}.
The equivalence of (2) -- (5) follows immediately from Lemma \ref{lemma-universally-injective-unramified}. Moreover, if (2) -- (5) are satisfied then $f$ is representable. Similarly, if (1) is satisfied then $f$ is representable. Hence the result follows from the case of schemes, see \'Etale Morphisms, Lemma \ref{etale-lemma-characterize-closed-immersion}.