Lemma 85.11.1. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of maps of sheaves on $(\mathit{Sch}/S)_{fppf}$. Assume (a) $X_ i$ is a formal algebraic space over $S$, (b) $X_ i \to X$ is representable by algebraic spaces and étale, and (c) $\coprod X_ i \to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$.

Proof. For each $i$ pick $\{ X_{ij} \to X_ i\} _{j \in J_ i}$ as in Definition 85.7.1. Then $\{ X_{ij} \to X\} _{i \in I, j \in J_ i}$ is a family as in Definition 85.7.1 for $X$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).