The Stacks project

Proof. We have seen above that the functor $F_ d$ corresponds to the open subscheme $U_ d \subset X$. Moreover the transformation of functors $F_ d \to F_{d'}$ (if $d | d'$) defined above corresponds to the inclusion morphism $U_ d \to U_{d'}$ (see discussion above). Hence to show that $F$ is represented by $X$ it suffices to show that $T \to X$ for a quasi-compact scheme $T$ ends up in some $U_ d$, and that for a general scheme $T$ we have

These verifications are omitted. $\square$