In Lee, there is an exercise involving the pullback that I can't understand. If $M,N$ are smooth manifolds and $F:M\to N$ is smooth, I am asked to show that the pullback $F^*$ is a smooth bundle map $T^*N \to T^*M$ where $T^*N$ denotes the cotangent bundle on $N$ and similarly for $M$.
The problem is that I can't figure out how this map is supposed to be defined and I can't find the definition in the book. For example, it doesn't seem clear what $F^*$ of a covector at a point not in the image of $F$ should be. Strangely enough I haven't been able to find the definition online either. Does anyone know the relevant definition?