**Proof.**
It is clear that (1) implies (2) and that (2) implies (3). Assume $\{ X_ i \to X\} $ and $X_ i \to Y_ i \to Y$ as in (3) and let a diagram as in (1) be given. Since $Y_ i \times _ Y V$ is a formal algebraic space (Lemma 85.11.2) we may pick coverings $\{ Y_{ij} \to Y_ i \times _ Y V\} $ as in Definition 85.7.1. For each $(i, j)$ we may similarly choose coverings $\{ X_{ijk} \to Y_{ij} \times _{Y_ i} X_ i \times _ X U\} $ as in Definition 85.7.1. Since $U$ is quasi-compact we can choose $(i_1, j_1, k_1), \ldots , (i_ n, j_ n, k_ n)$ such that

\[ X_{i_1 j_1 k_1} \amalg \ldots \amalg X_{i_ n j_ n k_ n} \longrightarrow U \]

is surjective. For $s = 1, \ldots , n$ consider the commutative diagram

\[ \xymatrix{ & & & X_{i_ s j_ s k_ s} \ar[ld] \ar[d] \ar[rd] \\ X \ar[d] & X_{i_ s} \ar[l] \ar[d] & X_{i_ s} \times _ X U \ar[l] \ar[d] & Y_{i_ s j_ s} \ar[ld] \ar[rd] & X_{i_ s} \times _ X U \ar[d] \ar[r] & U \ar[d] \ar[r] & X \ar[d] \\ Y & Y_{i_ s} \ar[l] & Y_{i_ s} \times _ Y V \ar[l] & & Y_{i_ s} \times _ Y V \ar[r] & V \ar[r] & Y } \]

Let us say that $P$ holds for a morphism of countably indexed affine formal algebraic spaces if it holds for the corresponding morphism of $\textit{WAdm}^{count}$. Observe that the maps $X_{i_ s j_ s k_ s} \to X_{i_ s}$, $Y_{i_ s j_ s} \to Y_{i_ s}$ are given by completions of étale ring maps, see Lemma 85.15.13. Hence we see that $P(X_{i_ s} \to Y_{i_ s})$ implies $P(X_{i_ s j_ s k_ s} \to Y_{i_ s j_ s})$ by axiom (1). Observe that the maps $Y_{i_ s j_ s} \to V$ are given by completions of étale rings maps (same lemma as before). By axiom (2) applied to the diagram

\[ \xymatrix{ X_{i_ s j_ s k_ s} \ar@{=}[r] \ar[d] & X_{i_ s j_ s k_ s} \ar[d] \\ Y_{i_ s j_ s} \ar[r] & V } \]

(this is permissible as identities are faithfully flat ring maps) we conclude that $P(X_{i_ s j_ s k_ s} \to V)$ holds. By axiom (3) we find that $P(\coprod _{s = 1, \ldots , n} X_{i_ s j_ s k_ s} \to V)$ holds. Since the morphism $\coprod X_{i_ s j_ s k_ s} \to U$ is surjective by construction, the corresponding morphism of $\textit{WAdm}^{count}$ is the completion of a faithfully flat étale ring map, see Lemma 85.15.14. One more application of axiom (2) (with $B' = B$) implies that $P(U \to V)$ is true as desired.
$\square$

## Comments (2)

Comment #1960 by Brian Conrad on

Comment #2014 by Johan on