Glueing cubes to manifolds with corner

Question

I am interested in proposition 3.7 in Salvatore's `Configuration spaces with summable labels'. The result states that the bar construction on the Fulton-Macpherson operad is isomorphic to the FM-operad itself. In his proof, the author claims that if you take the manifold with corners $F_n(i)$ and for each face $S$ of $F_n(i)$ of codimension $d$, glue a copy of $S \times [0,1]^d$, then this is diffeomorphic to the original manifold $F_n(i)$. I was wondering whether this is a general result to do with manifolds with corners and if so where can it be found in the literature?