The Stacks project

Proof. The representability of $F \to F \times F$ implies that each diagonal morphism $F_ i \to F_ i \times F_ i$ is representable (immediate from the definitions and the fact that $F \times _{(F \times F)} (F_ i \times F_ i) = F_ i$). Choose a scheme $U$ in $(\mathit{Sch}/S)_{fppf}$ and a surjective étale morphism $U \to F$ (this exist by hypothesis). The base change $U \times _ F F_ i \to F_ i$ is surjective and étale by Lemma 65.5.5. On the other hand, $U \times _ F F_ i$ is a scheme by Lemma 65.8.1. Thus we have verified all the conditions in Definition 65.6.1 and $F_ i$ is an algebraic space. $\square$