Smooth function on a manifold not dependent on coordinate chart

Question

I'm having trouble with the proof of the following remark from page 59 of Tu's book on Manifolds. The part I'm worried about is where he gets that $\phi\circ \psi^{-1}$ is $C^\infty$. Is he allowed to make that assumption? I thought we could only say this if the two charts were $C^\infty$-compatible.

Other relevant definitions from his book:

Answer 1

A smooth manifold is a $C^{\infty}$ manifold by definition and the transition functions of a $C^{\infty}$ manifold are supposed to be $C^{\infty}$ by definition.