Let $A$ be $\mathbb Z_{\geq 0}$ graded commutative ring. Now, in almost every resource I read, we define the structure a scheme on $\operatorname{Proj} A$, by taking the distinguished basis $D_+(f)$ of $\operatorname{Proj}A$, identifying $D_+(f)$ with $\operatorname{Spec}(A_f)_0$ and gluing the sheafs $\mathcal{O}_{\operatorname{Spec}(A_f)_0}$ together along the intersections. This makes sense to me, but why don't we just emulate the construction of the structure sheaf on $\operatorname{Spec}A$?
More precisely, let $\psi_f:\operatorname{Spec(A_f)_0}\rightarrow D_+(f)$ be the homemorphisms, and define a (pre)-sheaf on thee basis of distinguished opens by $D_+(f)\mapsto \mathcal{O}_{\operatorname{Spec(A_f)_0}}(\psi_f^{-1}(U_f))=(A_f)_0$. Why don't we just show this is a sheaf on the base of distinguished opens, and then define the structure sheaf on $\operatorname{Proj}A$ to be this sheaf?
I feel like these constructions should be equivalent, and I also feel like that the latter would be computationally less intensive, since we have already done the hard work of proving this is a sheaf on the base in the affine case? Maybe I am missing something though.