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I am reading Loring Tu's "Introduction to Manifolds" and I came across the following proposition:

Suppose $F:N \to M$ is $C^\infty$ at $p \in N$. If $(U, \phi)$ is any chart about $p$ in $N$ and $(V, \psi)$ is any chart about $F(p)$ in $M$, then $\psi \circ F \circ \phi^{-1}$ is $C^\infty$ at $\phi(p)$.

Proof. Since $F$ is $C^\infty$ at $p \in N$, there are charts $(U_\alpha, \phi_\alpha)$ about $p$ in $N$ and $(V_\beta, \psi_\beta)$ about $F(p)$ in $M$ such that $\psi_\beta \circ F \circ \phi_\alpha^{-1}$ is $C^\infty$ at $\phi_\alpha(p)$. By the $C^\infty$ compatibility of charts in a differentiable structure, both $\phi_\alpha \circ \phi$ and $\psi \circ \psi_\beta^{-1}$ on open subset of Euclidean spaces. Hence, the composite $$ \psi \circ F \circ \phi^{-1} = (\psi \circ \psi_\beta^{-1}) \circ (\psi_\beta \circ F \circ \phi_\alpha^{-1}) \circ (\phi_\alpha \circ \phi^{-1}) $$ is $C^\infty$ at $\phi(p)$.

What I don't understand is the reason $\phi$ and $\phi_\alpha$ (and also $\psi$ and $\psi_\beta$) should be compatible. Are all charts on a smooth manifold compatible? Or does the author mean any chart in the differentiable structure by the expression any chart?

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    $\begingroup$ Yes. When talking about charts of a differentiable manifold, they are implied to be taken from the differentiable structure. $\endgroup$ Sep 23, 2020 at 14:33

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Yes. Quotation from the end of section 5.3 (p. 53):

From now on, a “manifold” will mean a $C^\infty$-manifold. We use the terms “smooth” and $C^\infty$ interchangeably. [...] By a chart $(U,\phi)$ about $p$ in a manifold $M$, we will mean a chart in the differentiable structure of $M$ such that $p \in U$.

This means that the charts occurring in Definition 6.5 and Proposition 6.7 are tacitly assumed to belong to the fixed differentiable structure which determines $M$ as a smooth manifold. In particular, the charts $(U,\phi)$ and $(U_\alpha,\phi_\alpha)$ as well as the charts $(V,\psi)$ and $(V_\beta,\psi_\beta)$ are automatically compatible.

Note that the same applies for Definition 6.1 and Remark 6.2. See Smoothness of a function is independent from the chart .

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The author meant "any chart" in the differentiable structure (this is why they have to be compatible). In general, the maximal atlas/diff structure is eluded and charts are assumed to be of the structure.

Most likely you have seen a $C^{\infty}$ map defined along the lines "if there are charts such that...". This result helps bridge the gap between theory and practice: if you want to disprove a map is $C^{\infty}$, by the definition, you would have to check all charts (this is how you prove non-existence). However, after this result, you only have to find a pair of charts that don't make the local map be $C^{\infty}$.

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