Question:Let $S$ be a graded ring and $ f \in S_+$. Does the ring $ S_{(f)}$ which consists of degree $ 0$ elements of $ S_f$ represent a nice functor?

** Motivation: ** Let $ X = {\rm Spec} A$. Assume that $D(f) \subseteq D(g)$. It is messy to think about the restriction map $ A_g \to A_f $ using formulas. Things are much more transparent in the language of representable functors. Indeed, $A_g$ represents the functor

$$ R \mapsto \{ \text{maps $ A \to R$ which send $ g $ to a unit} \} $$

Since $ D(f) \subseteq D(g) $ we have a natural transformaion

$$ \{ \text{maps $ A \to R$ which send $ f $ to a unit} \} \subseteq \{ \text{maps $ A \to R$ which send $ g $ to a unit} \} $$

The yoneda lemma gives us a ring homomorphism $ A_g \to A_f $ which is the restriction map in the structure sheaf of the affine scheme. I just want to emphasize that I am not saying the yoneda lemma construction is better, it is just easier for me to keep in my head than a bunch of formulas.

Now let $S$ be a graded ring. We construct $ {\rm Proj} S $ by gluing together the affine schemes $ {\rm Spec} S_{(f)}$ where $ f \in S_{+}$ is homogeneous. In order to glue we need ring isomorphisms $$ S_{(fg)} \cong (S_{f})_{\frac{g^{\deg f}}{f^{\deg g}}} $$ If $ S_{(f)}$ represented a nice functor, we could try and mimic the construction of the restriction maps in the structure sheaf of an affine scheme. As it stands, the standard construction of this isomorphism is messy and very hard to remember (for me).