Dependence of smoothness of a real-valued function on a manifold

Loring Tu's An introduction to manifold, Proposition 6.3 asserts that:

Let $$M$$ be a smooth $$n$$-manifold and $$f:M \to \Bbb R$$ a real valued function on $$M$$. Then the following are equivalent:

(1) The function $$f:M \to \Bbb R$$ is $$C^{\infty}$$

(2) $$M$$ has an atlas such that for every chart $$(U,\varphi)$$ in the atlas $$f \circ \varphi ^{-1}$$ is $$C^{\infty}$$

But, I have a trouble with this. Since $$M$$ is a smooth $$n$$-manifold, it already has a smooth structure, and condition (1) is saying $$f$$ is smooth with respect to the structure of $$M$$. On the other hand, (2) is saying that $$M$$ has another atlas. Although the atlas in (2) is contained in a unique maximal atlas of $$M$$, it is not necessarily the same with the already given maximal atlas of $$M$$, isn't it?

The atlas in $$(2)$$ must be compatible with the smooth structure on $$M$$. What $$(2)$$ is saying is that you don't need to use a maximal atlas, just any atlas (it cuts down on the work needed to check things if there's an atlas with a finite number of charts).
The content in $$(2)$$ is that if $$f$$ is smooth with respect to an atlas $$\mathcal{A}$$, then it is also smooth with respect to the maximal atlas containing $$\mathcal{A}$$. The content is the extension to the charts of the maximal atlas not in $$\mathcal{A}$$.
For example, on a circle (with the standard atlas) you only need to check that a function $$f$$ is smooth with respect to the upper, lower, left, and right semicircles to conclude that $$f$$ is smooth instead of checking that its smooth with respect to all charts in a maximal atlas.