# Math Help - Hartshorne Question

1. ## Hartshorne Question

The functor $X \rightarrow A(X)$ induces an arrow-reversing equivalence of categories between the category of affine varieties over $k$ and the category of finitely generated integral domains over $k$.

My textbook (Hartshorne) says that this is a corollary to the following proposition:

Let $X$ be any variety and let $Y$ be an affine variety. Then there is a natural bijective mapping of sets

$\alpha : \text{Hom}(X, Y) \overset{\sim}{\rightarrow} \text{Hom}(A(Y), \mathcal{O}(X))$

where the left $\text{Hom}$ means morphisms of varieties, and the right $\text{Hom}$ means homomorphisms of $k$-algebras.

I do not see how to apply the proposition to prove this corollary. I would appreciate advice on this. Thanks.