Let $X$ be a Noetherian scheme (regular if needed), and let $\mathcal{E}$ be a locally free sheaf of rank $2$ on $X$. Let $\pi:\mathbb{P}(\mathcal{E})\to X$ be the natural morphism, and let $f:\mathcal{F}\to\mathcal{G}$ be a morphism of (quasi-)coherent $\mathcal{O}_{\mathbb{P}(\mathcal{E})}$-modules. Suppose fruthermore that for every $x\in X$, $f$ induces an isomorphism between the restrictions of $\mathcal{F}$ and $\mathcal{G}$ to the fiber over $x$. Is it true that then $f$ is an isomorphism? Can the conditions be relaxed, i.e. is it also true for a more general morphism of schemes $\pi':Y\to X$ and maybe more general sheaves of $\mathcal{O}_Y$ modules?
My attempt: denote for $x\in X$ by $F_x$ the fiber over $x$ and by $\iota_x:F_x\to \mathbb{P}(\mathcal{E})$ the natural morphism. By hypothesis, we have that $$ \iota_x^{*}f:\iota_x^{*}\mathcal{F}\to\iota_x^{*}\mathcal{G} $$ is an isomorphism for every $x$. Now let $y\in F_x$ be arbitrary. Now if my computations are correct (which is not a small hypothesis), then we have $$ \mathcal{O}_{F_x,y}=\mathcal{O}_{\mathbb{P}(\mathcal{E}),\iota_x(y)}/\mathfrak{m}_{X,x}^e $$ where $\mathfrak{m}_{X,x}^e$ is the extensions of $\mathfrak{m}_{X,x}$ under $\pi_{\iota_x(y)}^{\#}:\mathcal{O}_{X,x}\to \mathcal{O}_{\mathbb{P}(\mathcal{E}),\iota_x(y)}$. Also, we then have $$ (\iota_x^{*}\mathcal{F})_y=\mathcal{F}_{\iota_x(y)}/(\mathfrak{m}_{X,x}^e\cdot \mathcal{F}_{\iota_x(y)}) $$ and similarly for $\mathcal{G}$. Therefore, our hypothesis implies that the map $$ \bar{f}_{\iota_x(y)}:\mathcal{F}_{\iota_x(y)}/(\mathfrak{m}_{X,x}^e\cdot \mathcal{F}_{\iota_x(y)})\to \mathcal{G}_{\iota_x(y)}/(\mathfrak{m}_{X,x}^e\cdot \mathcal{G}_{\iota_x(y)}) $$ is an isomorphism. To conclude, we would have to show that then also $f_{\iota_x(y)}$ is an isomorphism, which is where I am currently stuck. Does this follow from e.g. Nakayamas Lemma?
Edit: I made some further progress: If we regard $\mathcal{F}_{\iota_x(y)}$ and $\mathcal{G}_{\iota_x(y)}$ as $\mathcal{O}_{X,x}$-modules via $\pi_{\iota_x(y)}^{\#}$, then Nakayamas lemma implies that $f_{\iota_x(y)}$ is surjective. For my pourposes this is enough, as I only need the statement for $\mathcal{F}$ and $\mathcal{G}$ locally free of the same finite rank, and then injectivity follows formally. But I still wonder if this hypothesis can be dropped.