Collar neighborhood theorem

Flow a global inward field from boundary points. In a boundary chart, extend the field across the face. The derivative of $(y,t)\mapsto {\mathrm{\Phi }}_t(y,0)$ at $t=0$ is the identity on face directions together with the inward vector in the time direction, hence is invertible. The Euclidean inverse function theorem therefore gives local collar embeddings.

A locally finite refinement of these local collars admits a smooth positive width $\delta$ for which $(p,t)\mapsto {\mathrm{\Phi }}_t(p)$ is injective on $0\le t<\delta (p)$ and has open image near the boundary; this is the usual locally finite shrinking of local collar domains.

The reparametrization $c(p,s)={\mathrm{\Phi }}_{s\delta (p)}(p)$ from $\mathrm{\partial }M\times [0,1)$ is then a smooth embedding, fixes $\mathrm{\partial }M$ at $s=0$, and has that open image. It is therefore a smooth collar.