Let Mbe a smooth manifold, and letp:E→M be a topological covering map. Then E has a smooth structure so that p becomes a smooth covering map. (This is shown in Proposition 4.40 of Lee's Introduction to Smooth Manifolds). Next I want to show that such a smooth structure of E is unique. How do I have to start to do this? I should suppose Ehas two different smooth structures and smooth covering maps p,q:E→M with respect to the two smooth structures, respectively, and then what I have to do? A smooth structure is by definition a maximal smooth atlas. Do I have to show that two atlases are the same by showing inclusions?