This is 4.2.2 in EGA II.
Let $q : X \rightarrow Y$ be a morphism of schemes. Let $L$ be an invertible sheaf on $X$ and let $E$ be a finite type quasicoherent sheaf on $Y$. Let $\phi : q^*(E) \rightarrow L$ be a surjective morphism. The next step is to get a corresponding morphism $S(q^*(E)) \rightarrow \oplus_{n \in \mathbb{N}} L^{\otimes n}$, which then corresponds to a morphism $X \rightarrow \mathbb{P}_Y(E)$
How is this morphism constructed? I'm thinking we can compose $\phi$ with the natural inclusion $L \rightarrow \oplus_{n \in \mathbb{N}} L^{\otimes n}$, and then use the universal property of symmetric algebras. But this requires that $x \otimes y = y \otimes x$ in $\oplus_{n \in \mathbb{N}} L^{\otimes n}$ for $x, y \in L(U)$. I'm not sure if this is true.