Bundle morphisms from maps of smooth sections ($\mathrm{Hom}(E,F)\cong {\mathrm{Hom}}_{C^{\infty }(M)}(\Gamma (E),\Gamma (F))$)

Every $C^{\infty }(M)$-linear map between the modules of smooth sections of finite-rank smooth vector bundles is induced by a unique vector bundle morphism. A smooth bump function shows that the map is local. A bumped local frame then proves that a section vanishing at $x$ has image vanishing at $x$. Evaluating the image therefore defines a well-defined fiberwise linear map. The images of the bumped frame give smooth matrix columns, proving the resulting morphism is smooth. This elementary argument is also described in Brian Conrad's bundle notes.