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. $\square$

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).