Below are excerpts from Lee's Introduction To Smooth Manifolds for the context of my question:
What I am confused about is the part where he talks about $\phi_i^+ \circ (\phi_i^-)^{-1} = \phi_i^- \circ (\phi_i^+)^{-1} = Id_{\mathbb{B}}$. There seems to be a mismatch of domain and range. We have $\phi_i^-: U_i^- \cap \mathbb{S}^n \rightarrow \mathbb{B}^n$ so $(\phi_i^-)^{-1}: \mathbb{B}^n \rightarrow U_i^- \cap \mathbb{S}^n$. But $\phi_i^+: U_i^+ \cap \mathbb{S}^n \rightarrow \mathbb{B}^n$ so I don't understand how $\phi_i^+ \circ (\phi_i^-)^{-1}$ makes sense.
Ugh. Example 1.31 is entirely messed up. Someone pointed this out to me several months ago, but I was too busy at the time and forgot to get back to it. I've now added a correction to my errata list.
Thanks for pointing it out.