I am reading Lee's book Introduction to Smooth Manifold. I am confused about the conception of pullback in this book.
Assume $F:M\to N$ is a smooth map. We can define a pullback $F^*$ at $p\in M$ associated with $F$ such as $$F^*:T^*_{F(p)}N\to T^*_pM$$ which is the dual map of tangent map $F^*$.
The local definition is very well. However, I am confused with the global one that occur in the exercise.
(Problem 6-2, P151) If $F:M\to N$ is a smooth map, show that $F^*:T^*N\to T^*M$ is a smooth bundle map.
I do not think that the local definition can be used in the global one directly.
$F^*$ is contravariant at each point, so it is not a bundle map.
Am I right? Any advice is helpful. Thank you.
Concerning the first point: maybe one should rather say that the pullback $$F^{\ast} (T^{\ast} N\to N) := T^{\ast} N\times_N M\to M$$ of the cotangent bundle $T^{\ast}N\to N$ along $F: M\to N$ admits a natural bundle morphism to the cotangent bundle $T^{\ast}M\to M$ of $M$, which is fiberwise given by the pullback map $T^{\ast}_{F(p)}N\to T^{\ast}_p M$ you described.
To check smoothness, you may assume that $M$ and $N$ are both affine spaces, since any point of $M$ admits a chart which is mapped into a chart of $N$.