In the definition of F-related vector fields, F must be a diffeomorphism. Why must it be a diffeomorphism? What if F is smooth and bijective, but not a diffeo?
The definition does not even require F-->N to be bijective. If F is not bijective, then a vector field on M might to push-foward to a vector field on N, and a vector field on N might not have an F-related vector field on M. But if F is a diffeo, then we are in the nice situation where to every vector field on N, then exists a unique F-related vector field on M given, of course, by the pushfoward by F^-1.