Let $\pi_E:E\to M$ and $\pi_F:F\to M$ be smooth real vector bundles over a smooth manifold $M$. Let $\operatorname{Hom}(E,F)\to M$ denote the smooth vector bundle whose fibre over $p\in M$ is $\operatorname{Hom}(E_p,F_p)$. Then there is a natural bijection
where $\operatorname{Hom}_M(E,F)$ is the set of smooth vector bundle maps $\Phi:E\to F$ covering $\operatorname{id}_M$, meaning $\pi_F\circ \Phi=\pi_E$ and each fibre restriction $\Phi|_{E_p}:E_p\to F_p$ is linear.
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Under this bijection, a section $A\in \Gamma(M,\operatorname{Hom}(E,F))$ corresponds to the bundle map $\Phi_A:E\to F$ defined by