My understanding is that the forgetful functor from the category of varieties, as defined in Hartshorne Chapter 1 for example, to Set is faithful. That is to say, morphisms are maps of sets with particular special properties.
Since there is a fully faithful functor from varieites to Sch whose essential image comprises the subcategory of abstract varieties, it seems that it would follow that the forgetful functor from the category of abstract varieties to Set is also faithful.
I am under the impression that the forgetful functor from Sch to Set is not faithful, however, since one must specify the morphism of sheaves in addition to the topological map.
Is what I have said thus far correct? If so, my question is: how exactly are the morphisms of sheaves determined by the topological morphisms when we look in the subcategory of abstract varieties? Of course classically one can just pull back regular functions—but how does this work scheme-theoretically?