I have a minor problem on understanding the definition. Say, in the book An Introduction of Manifolds by Loring Tu, he gave a definition as below. My question is: are the chart $(U,\phi)$ and $(V,\psi)$ mentioned in the definition belongs to the atlas of the given manifold $N$ and $M$? To be clear, we know that when we say "manifold $N$", we mean a underlying set together with a "maximal atlas $\Phi_N$," so is $\phi\in\Phi_N$ here? It's hard to guess from the context.
Also, Prop 6.8(ii)(iii) is weird to me:
Yes: if $M$ is a smooth manifold, then whenever you talk about a chart in $M$ (or on $M$, or of $M$, etc.) that always refers to a chart in the atlas of $M$ unless specified otherwise.