Lemma 87.34.6. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{spaces, {\acute{e}tale}}$ is equivalent to the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $\varphi $ is representable by algebraic spaces and étale.
Proof. Denote $\mathcal{C}$ the category introduced in the lemma. Recall that $X_{spaces, {\acute{e}tale}} = X_{red, spaces, {\acute{e}tale}}$. Hence we can define a functor
\[ \mathcal{C} \longrightarrow X_{spaces, {\acute{e}tale}},\quad (U \to X) \longmapsto U \times _ X X_{red} \]
because $U \times _ X X_{red}$ is an algebraic space étale over $X_{red}$.
To finish the proof we will construct a quasi-inverse. Choose an object $\psi : V \to X_{red}$ of $X_{red, spaces, {\acute{e}tale}}$. Consider the functor $U_{V, \psi } : (\mathit{Sch}/S)_{fppf} \to \textit{Sets}$ given by
\[ U_{V, \psi }(T) = \{ (a, b) \mid a : T \to X, \ b : T \times _{a, X} X_{red} \to V, \ \psi \circ b = a|_{T \times _{a, X} X_{red}}\} \]
We claim that the transformation $U_{V, \psi } \to X$, $(a, b) \mapsto a$ defines an object of the category $\mathcal{C}$. First, let's prove that $U_{V, \psi }$ is a formal algebraic space. Observe that $U_{V, \psi }$ is a sheaf for the fppf topology (some details omitted). Next, suppose that $X_ i \to X$ is an étale covering by affine formal algebraic spaces as in Definition 87.11.1. Set $V_ i = V \times _{X_{red}} X_{i, red}$ and denote $\psi _ i : V_ i \to X_{i, red}$ the projection. Then we have
\[ U_{V, \psi } \times _ X X_ i = U_{V_ i, \psi _ i} \]
by a formal argument because $X_{i, red} = X_ i \times _ X X_{red}$ (as $X_ i \to X$ is representable by algebraic spaces and étale). Hence it suffices to show that $U_{V_ i, \psi _ i}$ is an affine formal algebraic space, because then we will have a covering $U_{V_ i, \psi _ i} \to U_{V, \psi }$ as in Definition 87.11.1. On the other hand, we have seen in the proof of Lemma 87.34.3 that $\psi _ i : V_ i \to X_ i$ is the base change of a representable and étale morphism $U_ i \to X_ i$ of affine formal algebraic spaces. Then it is not hard to see that $U_ i = U_{V_ i, \psi _ i}$ as desired.
We omit the verification that $U_{V, \psi } \to X$ is representable by algebraic spaces and étale. Thus we obtain our functor $(V, \psi ) \mapsto (U_{V, \psi } \to X)$ in the other direction. We omit the verification that the constructions are mutually inverse to each other. $\square$
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