Bundle morphisms in terms of sections

Let $$E$$ and $$F$$ be complex vector bundles on a smooth manifold $$M$$. I'm trying to prove that the $$C^{\infty}(M)-$$module morphisms $$L: \Gamma(E) \rightarrow \Gamma(F)$$ are in correspondence with the bundle morphisms $$\tilde{L}: E \rightarrow F$$. But I can't see how to construct such a bundle morphism from a $$C^{\infty}(M)-$$module morphism $$L$$.

My attempt so far: Given $$e \in E$$, let's choose a local smooth section $$s: U \subset M \rightarrow E$$ defined on a neighbourhood of $$\pi_E(e)$$ because $$\pi_E$$ is a submersion. Also, let's choose a bump function $$\chi \in C^{\infty}(M)$$ with compact support, and with the property that there exists a neighbourhood of $$\pi_E(e)$$ on which $$\chi \equiv 1$$. Then, $$\chi s$$ has compact support, and we can extend it trivially to all of $$M$$. Let's define $$\tilde{L}(e) = L(\chi s)(\pi_E(e)).$$

I proved that this definition is good after fixing $$s$$, but I couldn't prove that it is independent of the choice of s.

Thank you in advance for any help.

• Hint: use a local frame. Jun 7, 2023 at 23:22