Lemma 109.55.1. There exists a functor $F : \mathit{Sch}^{opp} \to \textit{Sets}$ which satisfies the sheaf condition for the fpqc topology, has representable diagonal $\Delta : F \to F \times F$, and such that there exists a surjective, flat, universally open, quasi-compact morphism $U \to F$ where $U$ is a scheme, but such that $F$ is not an algebraic space.

**Proof.**
See discussion above.
$\square$

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