The Stacks project

Proof. Recall that the sites $(\mathit{Sch}/S)_{fppf}$ and $(\mathit{Sch}/S)_{\acute{e}tale}$ have the same underlying category, see discussion in Section 98.25. Similarly the sites $(\textit{Noetherian}/S)_{\acute{e}tale}$ and $(\textit{Noetherian}/S)_{fppf}$ have the same underlying categories. By axioms [0] and [1] the functor $F$ is a sheaf and limit preserving. Let $F' : (\mathit{Sch}/S)_{\acute{e}tale}^{opp} \to \textit{Sets}$ be the unique extension of $F$ which is a sheaf (for the étale topology) and which is limit preserving, see Lemma 98.25.2. Then $F'$ satisfies axioms [0] and [1] as given in Section 98.15. By Lemma 98.25.4 we see that $\Delta ' : F' \to F' \times F'$ is representable (by schemes). On the other hand, it is immediately clear that $F'$ satisfies axioms [-1], [2], [3], [4], [5] of Section 98.15 as each of these involves only evaluating $F'$ at objects of $(\textit{Noetherian}/S)_{\acute{e}tale}$ and we've assumed the corresponding conditions for $F$. Whence $F'$ is an algebraic space by Proposition 98.16.1. $\square$