I am looking at the proof of Proposition 5.2 from John Lee's ITM. It says below that the uniqueness of smooth structure part of the proposition is obvious. I wasn't sure how to prove this obvious statement. After some thought, I think the formal argument is as follows. Let $\tilde{S}$ be the set $S$ endowed with another smooth structure making it into an embedded submanifold of $M$ with the property that $F$ is a diffeomorphism onto its image. Then $S$ and $\tilde{S}$ have the same smooth structure if and only if the identity map between the two sets is diffeomorphic. Say this map is $I_S$. Then $I_S = F \circ I_N \circ F|_S^{-1}$ where $I_N$ is the identity map for $N$. Since all three maps are diffeomorphisms, so is $I_S$.
Is this argument correct?
