Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ where $h: \partial \mathbb{D}^{\lambda} \times \mathbb{D}^{\mu} \rightarrow \partial M$ is a smooth imbedding. This leaves us with a smooth manifold with corners. The only method I have read about for smoothing the corners has the disadvantage that $M$ is not a smooth submanifold of the resulting smooth manifold with boundary, but merely a smooth submanifold with corners. I find this very irksome and I was wondering if anyone knows whether there is a smooth structure on the quotient $M' := M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ so that the inclusion $M \rightarrow M'$ is a smooth imbedding. In the picture I have in my head, M is not altered at all, and the corners of the cylinder, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, curve to meet the boundary of $M$. This picture gives me hope that such a smoothing is possible, however the only method I have learned for endowing the quotient with a smooth structure involves a homeomorphism of $M$ with a submanifold (with corners) of itself.
Thank you for your time.