In this proof we redo some of the arguments used in the proofs of Lemmas \ref{lemma-first-order-thickening-surjective} and \ref{lemma-thickening-scheme}. We first handle the case $B = S = \Spec(\mathbf{Z})$. Let $U$ be an affine scheme, and let $U \to X$ be \'etale. Then $$ 0 \to \mathcal{I}(U) \to \mathcal{A}(U) \to \mathcal{O}_X(U) \to 0 $$ is exact as $H^1(U_\etale, \mathcal{I}) = 0$ as $\mathcal{I}$ is quasi-coherent, see Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. If $V \to U$ is a morphism of affine objects of $X_{spaces, \etale}$ then $$ \mathcal{I}(V) = \mathcal{I}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{O}_X(V) $$ since $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_X$-module, see Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}. Hence $\mathcal{A}(U) \to \mathcal{A}(V)$ is an \'etale ring map, see Algebra, Lemma \ref{algebra-lemma-lift-etale-infinitesimal}. Hence we see that $$ U \longmapsto U' = \Spec(\mathcal{A}(U)) $$ is a functor from $X_{affine, \etale}$ to the category of affine schemes and \'etale morphisms. In fact, we claim that this functor can be extended to a functor $U \mapsto U'$ on all of $X_\etale$. To see this, if $U$ is an object of $X_\etale$, note that $$ 0 \to \mathcal{I}|_{U_{Zar}} \to \mathcal{A}|_{U_{Zar}} \to \mathcal{O}_X|_{U_{Zar}} \to 0 $$ and $\mathcal{I}|_{U_{Zar}}$ is a quasi-coherent sheaf on $U$, see Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent-functorial}. Hence by More on Morphisms, Lemma \ref{more-morphisms-lemma-first-order-thickening} we obtain a first order thickening $U \subset U'$ of schemes such that $\mathcal{O}_{U'}$ is isomorphic to $\mathcal{A}|_{U_{Zar}}$. It is clear that this construction is compatible with the construction for affines above. \medskip\noindent Choose a presentation $X = U/R$, see Spaces, Definition \ref{spaces-definition-presentation} so that $s, t : R \to U$ define an \'etale equivalence relation. Applying the functor above we obtain an \'etale equivalence relation $s', t' : R' \to U'$ in schemes. Consider the algebraic space $X' = U'/R'$ (see Spaces, Theorem \ref{spaces-theorem-presentation}). The morphism $X = U/R \to U'/R' = X'$ is a first order thickening. Consider $\mathcal{O}_{X'}$ viewed as a sheaf on $X_\etale$. By construction we have an isomorphism $$ \gamma : \mathcal{O}_{X'}|_{U_\etale} \longrightarrow \mathcal{A}|_{U_\etale} $$ such that $s^{-1}\gamma$ agrees with $t^{-1}\gamma$ on $R_\etale$. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-descent-sheaf} this implies that $\gamma$ comes from a unique isomorphism $\mathcal{O}_{X'} \to \mathcal{A}$ as desired. \medskip\noindent To handle the case of a general base algebraic space $B$, we first construct $X'$ as an algebraic space over $\mathbf{Z}$ as above. Then we use the isomorphism $\mathcal{O}_{X'} \to \mathcal{A}$ to define $f^{-1}\mathcal{O}_B \to \mathcal{O}_{X'}$. According to Lemma \ref{lemma-first-order-thickening-maps} this defines a morphism $X' \to B$ compatible with the given morphism $X \to B$ and we are done.