Suppose that we have a functor $$F:\operatorname{Sch} \to \operatorname{Set}$$ from the categorie of schemes to the category of sets. And suppose that when we restrict it to $\operatorname{AffSch}$ the category of affine schemes we obtain that this functor is representable. What else do we need to check in order to be able to say that $F$ is itself representable?
It seems to me that the representing object that we get when we restrict the functor should tell you the affine pieces you need to glue to form the global representing functor. However I don't know how to make this precise.