Vector space of sections of dual to covariant bundle is contravariant

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Suppose F is a covariant functor on the category of differential manifolds with smooth maps, that associates to every differential manifold, a vector bundle over it, in a covariant fashion (so that a smooth map of differential manifolds induces a smooth map of the vector bundles over them). Then, we can define the following contravariant functor:

This functor sends any differential manifold M to the vector space of sections of the dual vector bundle to F(M).

Note that we need to do both operations: take the dual bundle, and then take the vector space of sections, to get a contravariant functor.