I am working on exercise II.7.10(c) in Hartshorne's Algebraic geometry, which asks:
Let $X$ be a noetherian regular scheme. Show that every $\mathbb{P}^n$-bundle $P$ over $X$ is isomorphic to $\mathbb{P}(\mathscr{E})$ for some locally free sheaf $\mathscr{E}$ on $X$.
Hartshorne gives an outline for this problem: Let $U \subseteq X$ be an open set such that $\pi^{-1}(U) \cong U \times \mathbb{P}^n$, and extend the invertible sheaf $\mathscr{O}(1)$ on $U \times \mathbb{P}^n$ to an invertible sheaf $\mathscr{L}$ on $P$. Set $\mathscr{E} = \pi_*\mathscr{L}$; then $P \cong \mathbb{P}(\mathscr{E})$.
I have some questions about showing the last isomorphism. At first, I thought there was a slick way to show that $P \cong \mathbb{P}(\mathscr{E})$, but now I'm not so sure. The argument begins by using proposition 7.12, which says that giving a morphism $P \to \mathbb{P}(\mathscr{E})$ is the same as giving an invertible sheaf $\mathscr{L}$ on $P$ and a surjective map $\pi^*\mathscr{E} \to \mathscr{L}$ of sheaves on $P$.
I originally thought that if the map $\pi^*\mathscr{E} \to \mathscr{L}$ were actually an isomorphism, the the map $P \to \mathbb{P}(\mathscr{E})$ I get is also an isomorphism. So my first question is: Is this true?
Secondly, since $\mathscr{E} = \pi_*\mathscr{L}$, I thought to use the adjunction \begin{equation*} \operatorname{Hom}(\pi_*\mathscr{L}, \pi_*\mathscr{L}) = \operatorname{Hom}(\pi^*\pi_*\mathscr{L},\mathscr{L}) \end{equation*} to get my isomorphism, namely the counit $\epsilon(\mathscr{L})$. But then I realized that even if the identity map $\pi_*\mathscr{L} \to \pi_*\mathscr{L}$ is an isomorphism, it is not necessarily the case that isomorphisms are preserved by adjunctions. In fact, I recently learnt that there is a special term for this. So my second question: Are my assumptions strong enough to guarantee that $\epsilon(\mathscr{L})$ is an isomorphism? (Maybe this SE question will be helpful.)
Some more context for my confusion: when I googled this problem, several solutions followed the same approach for this step, so I guess that this is evidence that the claims above are true, but whose proofs currently elude me. On the other hand, it was pointed out to me by someone else that $\pi^*\pi_*\mathscr{L}$ has rank $n+1$, and so cannot be isomorphic to $\mathscr{L}$, which answers my second question in the negative. Can someone please clarify the conflicting information?