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Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x \simeq \mathcal{F}(x)$, where with $\mathcal{F}(x)$ I mean the fibre of the sheaf over the point $x$. We also know that the sheaf of sections of the vector bundle $F$ is isomorphic to the locally free sheaf we started with. My question is the following: a section of $F$ is by definition a regular map $\sigma : X \rightarrow F$ such that $\sigma(x) \in F_x$ for any $x \in X$, whereas a section of $\mathcal{F}$ can be interpreted as a regular map $\psi : X \rightarrow \sqcup_{x \in X} \mathcal{F}_x$. I can see a natural map from “sections of $\mathcal{F}$” to “sections of $F$”, but how can one claim this is an isomorphism? Am I mixing something up?

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