I was reading a bit about the functor of points of a scheme, and it was mentioned that there does not exist a scheme $Y$ so that Hom$(Y,X)$ determines the points $X$ for all schemes $X$. This is in contrast to a group whose set $|G|$ is given by $|G| = Hom(\mathbb{Z},G)$, for example.
Is there a simple example/proof no such object $Y$ could exist?
If I understand correctly, you want to show that the forgetful functor from schemes to sets is not representable.
Suppose $Y$ represents this functor. Then $Hom_{sch}(Y,Spec(\mathbb{Z}))$ is infinite since $Spec(\mathbb{Z})$ has an infinite underlying set. But $Spec(\mathbb{Z})$ is a final object in the category of schemes : ther is exactly one morphism $Y\to Spec(\mathbb{Z})$. Contradiction.