Y-Embedding of a Scheme

Question

Let $f:X \to Y$ a morphism of schemes and $\mathcal{F}$ a $\mathcal{O}_Y$ module. Futhermore we have a embedding $X \subset Proj(Sym^{\cdot} (\mathcal{F}))$.

My question is what does it mean if $X \subset Proj(Sym^{\cdot} (\mathcal{F}))$ is a "$Y$-embedding"? Therefore what extra structure induces the "$Y$-blabla?

Answer 1

This means that $X \subset Proj(Sym^{\cdot} (\mathcal{F})$ is actually a commutative triangle (since both schemes come with a fixed map to $Y$). The extra structure comes from the fact that the relevant sheaves are $\mathcal O_Y$-modules.