Vector space of sections of dual to covariant bundle is contravariant

Statement

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.