Vector space of sections of dual to covariant bundle is contravariant
Suppose 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 to the vector space of sections of the dual vector bundle to .
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.