The following is Theorem 9.29 in Lee's Introducton to Smooth Manifolds.
Theorem 9.29. Let $M$ and $N$ be smooth $n$-manifolds with nonempty boundaries, and suppose $h:\partial N\to \partial M$ 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 $M',N'\subset M\cup_h N$ diffeomorphic to $M$ and $N$, respectively, and satisfying $M'\cup N'=M\cup_h N$ and $M'\cap N'=\partial M'=\partial N'$.
I want to know: Which functions on $M\cup_h N$ are smooth? In the topological category, a function on $M\cup_h N$ is continuous if and only if it is continuous on $M'$ and on $N'$. But I don't expect that this hold in the smooth category.
(Actuall my ultimate goal is to prove that the operation between cobordism classes is well-defined (p.3 (of the book not the pdf) in https://www.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf. But to this end it seems I need an answer of the question above.) Thanks in advance.