Let $M$ and $N$ be two manifolds with boundary and let $$f, g:\partial N\longrightarrow \partial M,$$ two isotopic diffeomorphisms, that is, there exists a diffeomorphism $$F:\partial N\times [0, 1]\longrightarrow \partial M$$ such that $$F(x, 0)=f(x)\quad \textrm{and}\quad F(x, 1)=g(x).$$ It is easy to show $g^{-1}\circ f:\partial N\longrightarrow \partial N$ is isotopic to $id_{\partial N}$ for it suffices considering the isotopy $$H(x, t)=(g^{-1}\circ F)(x, t).$$ How can I show $g^{-1}\circ f$ extends to a diffeomorphism $h:N\longrightarrow N$?
Obs: Maybe the collar theorem will be useful. That theorem asserts there exists a diffeomorphism $$\alpha:\partial N\times [0, 1)\longrightarrow N,$$ such that $\alpha|_{\partial N\times \{0\}}=id_{\partial N}$.