I have a question about a result in Lee's Introduction to Smooth Manifold(2nd Edition).
Theorem 9.29 (Attaching Smooth Manifolds Along Their Boundaries). Let $M$ and $N$ be smooth $n$-manifolds with nonempty boundaries, and suppose $h:\partial M \rightarrow \partial N$ is a diffeomorphism. Let$M \cup_h N $ be the adjunction space formed by identifying each $x \in \partial N$ with $h(x)\in \partial M$. Then $M \cup_h N $ is a topological manifold (without boundary), and has a smooth structure such that there are regular domains $M', N' $ diffeomorphic to $M ,N$, respectively, and satisfying $M'\cup N'=M \cup_h N $, $M'\cap N'=\partial N'=\partial M'$
In the proof of Theorem 9.29 we construct a smooth structure on the glued quotient space $M \cup_h N $. In such construction we take a collar neighborhood of $\partial M$ and $\partial N$ respectively, identify the union of them in the quotient space with $\partial M \times (-1,1)$. And the atlas on $M \cup_h N $ is the union of the atlas on $IntM, IntN$ respectively, and the atlas on $\partial M \times (-1,1)$.
Theorem 9.29 only claim the existence of such smooth structure. However from a following example (the double of a smooth manifold with boundary) it seems that the author assumes that we can get a canonical smooth structure on $M \cup_h N $ once $M,N$ and $h$ are given, independent of the choice of the collar neighborhoods. I'm thus wondering whether we have uniqueness from the setting of Theorem 9.29 (only $M,N$ and $h$ are fixed), and how to prove such uniqueness(if there exist).