= Bundle morphisms from maps of smooth sections
{title2=$\operatorname{Hom}(E,F)\cong\operatorname{Hom}_{C^\infty(M)}(\Gamma(E),\Gamma(F))$}
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 https://math.stanford.edu/~conrad/diffgeomPage/handouts/bundle.pdf[Brian Conrad's bundle notes].
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