I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it.
Let $M^m$ be a smooth manifold with boundary. We may form a new topological manifold by "adjoining a handle to $M$", I.e. by choosing a smooth map $h : \partial \mathbb{D}^{\mu} \times \mathbb{D}^{\lambda} \rightarrow \partial M$ ($\mu + \lambda = m$) and taking our new manifold to be $M' := M \cup_h \mathbb{D}^{\mu} \times \mathbb{D}^{\lambda}$.
A priori $M'$ is just a topological manifold with boundary, however there are many well-known ways to "smooth the corners" of $M'$ (I.e. to endow $M'$ with a smooth structure). My question is "what is the right uniqueness statement?" i.e. how can I formally state that all off these different ways to smooth corners result in diffeomorphic manifolds.
To give a sense for the sort of result I'm hoping for I'll say that I have in mind a statement that begins something like "up to diffeomorphism there is a unique smooth atlas for $M'$ such that the inclusion $M \subset M'$ is a smooth imbedding and the inclusion $\mathbb{D}^{\mu} \times \mathbb{D}^{\lambda} \subset M'$ is a smooth imbedding of a manifold with corners." That statement is probably nonsense but I just wanted to give a sense for the sort of thing I'd like to be true. Thanks for considering my question.
