In John Lee's Introduction to Smooth Manifolds, he states a characteristic property of surjective smooth submersions as for quotient maps. He refers to this "characteristic" property in Problem 4-7, which initially states that it is the unique smooth structure that satisfies a certain property. But then the problem actually asks to show that for any two manifolds satisfying the property, the two manifolds are diffeomorphic. The precise statement is given below.
However, I am a little confused with this. Because the book has a paragraph early on which states that diffeomorphism is different from having the same smooth structure (e.g. the standard structure on $\mathbb{R}$, and the smooth structure generated by $\psi(x)=x^{1/3}$ on $\mathbb{R}$ are diffeomorphic but distinct structures). So for the characteristic property referred to below, is it simply that any two such manifolds are diffeomorphic or they should actually have the same smooth manifold structure?
Suppose $M$ and $N$ are smooth manifolds, and $\pi:M \to N$ is a surjective smooth submersion. Show that there is no other smooth manifold structure on $N$ that satisfies the conclusion of Theorem 4.29; in other words, assuming that $\tilde{N}$ represents the same set as $N$ with a possibly different topoogy and smooth structure, and that for every smooth manifold $P$ with or without boundary, a map $F:\tilde{N} \to P$ is smooth if and only if $F \circ \pi$ is smooth, show that I$d_N$ is a diffeomorphism between $N$ and $\tilde{N}$.