Let $M$ be a smooth manifold with boundary $\partial M$. Let $Diff_0(M)$ be the group of all diffeomorphisms homotopic to identity. According to this article (Page 6, section " Beyond mapping class group"), the restriction of a diffeomorphism to the boundary gives a well defined surjactive homomorphism $$\phi: Diff_0(M)\rightarrow Diff_0(\partial M).$$
I could not find a reference for the above two results. So I have two questions.
1) Why is $\phi$ well defined, i.e., why the restriction of a difeomorphism homotopic to identity in $M$ is homotopic to identity in $\partial M$.
2) Why every diffeomorphism homotopic to identity in $\partial M$ is a restriction of a diffeomorphism of $M$ which is homotopic to identity.
As I am not quite familiar with diffeomorphism groups, any suggestion/reference/comment will be extremely helpful. Also I would request you to improve the tags if possible.
Thanks in advance.