If are two affine varieties, then and are isomorphic if and only if and are isomorphic as -algebras.
In my textbook (Hartshorne), it says that this immediately follows from a proposition that states:
Let be any variety and let be an affine variety. Then there is a natural bijective mapping of sets
where the left means morphisms of varieties, and the right means homomorphisms of -algebras.
How is the proof of this immediate from the proposition? I do not see why. Thanks.