I'm a little confused. While it is true that the cotangent space of a manifold naturally carries the structure of a smooth manifold (it's just a finite dimensional vector space) this is a strange question. I believe what you meant to ask is whether or not a diffeomorphism induces a linear isomorphism on the cotangent spaces. This is in fact true. In particular, let's suppose that we have a diffeomorphism . I think you know then that the derivative induces a linear isomorphism for each point . Dualizing this, you get an isomorphism , which is the desired isomorphism.