Charts in manifold

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.
• The equivalence of (ii) and (iii) is not trivial. (iii) refers to every chart in the atlas of definition of $M$ (i.e., the maximal atlas) while (ii) refers to only charts in some atlas (which may be a proper subset of the maximal atlas). Jul 13, 2019 at 17:50