Suppose $S$ is a finitely generated $A$-algebra. I'd like to describe a natural morphism between $\operatorname{Proj}S$ and $\operatorname{Spec}A$.
In exercise 6.3.F in Vakil's AG notes, we show that morphisms $X\rightarrow\operatorname{Spec}A$ are in natural bijections with ring morphisms $A\rightarrow\Gamma(\mathcal{O}_X,X)$.
Why does this result gives us the desired natural morphism?