I am trying to understand projective morphisms (primarily from reading Ravi Vakil's Foundations of Algebraic Geometry notes Ch 17 and 18) and I have run across a very basic problem. Say I have a ring $A,$ a graded $A$ algebras $S$ which is finitely generated in degree 1, an $A$-scheme $Y$ with structure morphism $\pi:Y\rightarrow \text{spec} A,$ and the structure morphism $\beta: \text{Proj} S\rightarrow \text{spec} A.$
In general it is true that maps of $A$-schemes, $f: Y\rightarrow \text{Proj} S$ are in correspondence with maps of graded rings $t_1:S\rightarrow \Gamma_{\ast}(X,L)$ for some line bundle $L$ on $Y$ where $L$ is globally generated by $f(S_1)$. (This is 17.4.2).
However later in the notes 18.2.1, Ravi states that such maps should be in bijection with maps $t_2:S\rightarrow \oplus_{n\geq 0} \pi_{\ast}L^{\otimes n}$ where there is a similar condition on being surjective in degree 1 and in this case I am viewing $S$ as an $O_{\text{spec} A}$ module.
What I don't understand is how maps $t_1$ correspond to maps $t_2$ aka why maps on global sections correspond to maps of sheaves. Morally speaking $L = f_{\ast} O(1)$ so if we knew that $f_{\ast}f^{\ast} O(1)$ was quasicoherent on $\text{Proj} S$ then using the fact that $\pi_{\ast} L = \beta_{\ast}f_{\ast}f^{\ast} O(1)$ and the fact that $\beta$ is qcqs (it is proper and $\text{Proj} S$ is quasicompact) would imply that $\pi_{\ast} L$ is quasicoherent. However I am not sure how to prove this result or whether I am looking at things the right way. Any guidance would be appreciated and please feel free to ask me to clarify.