I've just read a proof that
If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that the obvious maps $M \rightarrow M \cup_f N$ and $N \rightarrow M \cup_f N$ are smooth imbeddings.
The proof I read uses collar neighborhoods of the two boundaries to identify a neighborhood of the common boundary in the new manifold with a product of the common boundary and an interval.
This left me wondering about the uniqueness of the smooth structure. At first I thought it must be unique and I tried to show that the identity map is smooth but I couldn't show smoothness at points on the common boundary. Then the thought of a decomposition of an exotic sphere into hemispheres made me think perhaps uniqueness isn't guaranteed. But then I wasn't sure whether the hemispheres were still $smooth$ submanifolds when you change to the exotic smooth structure. Can anyone help me out by telling me whether we always have uniqueness and if so is it easy to see that the identity map will be smooth at points in the common boundary of M and N? Thanks very much for your time.