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Let $S$ be a nice base scheme and let $F : \mathsf{Sch}/S \to \mathsf{Set}$ be a functor. Are there necessary and sufficient conditions that $F$ is represented by some scheme $X$, i.e. $F \cong \hom(X,-)$? An obvious necessary condition is that $F$ is continuous. On affine schemes this would already suffice, but $F$ is not determined by what it does on affine schemes.

The case of functors $(\mathsf{Sch}/S)^{\mathrm{op}} \to \mathsf{Set}$ is studied a lot in the literature. For these there is also the notion of corepresenting schemes (besides the usual representing schemes), which should not be confused with the condition above.

Bonus question: Is there some geometric reconstruction of $X$ from the functor $F$?

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