I have a last question about the book by Wedhorn/Görtz (Algebraic Geometry I: Schemes), chapter 11 (on vector bundles).
They define first the quasi coherent bundle defined by $\mathcal{E}$ as follows: (see definition 11.2 and proposition 11.3):
The objective is now to prove an equivalence between geometric vector bundles and locally free $\mathcal{O}_X$ modules of rank $n$ (see definition 11.5 for how they define a geometric vector bundle over a scheme $X$):
I don't quite understand how from the reasoning above, it follows that $\mathbb{V} (\mathcal{E})$ is a geometric bundle of rank $n$? I understand that we have these isomorphisms $c: \mathcal{E}_{|U} \xrightarrow{\sim} (\mathcal{O}_{U}^{n})^{\vee}$. Does he now apply the fact from earlier, i.e. that $\text{Hom}_{\mathcal{O}_X} (\mathcal{F}, \mathcal{E}) \rightarrow \text{Hom}_X (\mathbb{V} (\mathcal{E}), \mathbb{V} (\mathcal{F}))$ is injective? i.e. does he apply the functor $\mathbb{V}$ to the above isomorphisms?
Here is also the converse direction:
I understand that we may assume that $V = \mathbb{V} (( \mathcal{O}_X^n)^{\vee})$. However, I don't understand how from proposition 11.3 it follows that then $\mathscr{S}(V/X) = \mathcal{O}_X^{n}$??
Also, how can one prove the equation (11.4.1), i.e. that $$ f^{*} \mathscr{S} (V/X) = \mathscr{S} ((V \times_X X^{'})/ X^{'}) $$