In Lee's Introduction to smooth manifolds I've come across this proof.
It is unclear to me why we need to obtain this new chart $(U_0, \varphi)$, rather than taking any chart $(W, \gamma)$ of $N$ containing $p$, from the very beginning, say, and just show that $\tilde{\psi} \circ F \circ \gamma^{-1}$ is smooth as a map from $\gamma(W \cap V_0) $ to $\psi(W \cap V_0)$ ?