# Sections of a locally free sheaf

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?