I am studying the proof that smooth, connected, 1-dimensional manifold are diffeomorphic to $\mathbb{R}$ or some type of interval given in Milnor's book "Topology From the Differentiable Viewpoint". I'm stuck in the art where he is building a diffeomorphism between $S^1$ and $M$, more specifically right when he defines this $h$ function and claims that this is a diffeomorphism, first of all I can't see that this is a homeomorphism because the domain is longer than $(0,2\pi)$. I found a similar, if not equal, proof written slightly different here but again this is very confusing to me. With the idea of quotient it feels a bit clearer but still, the diffeomorphism is not obvious.
I really appreciate any help. (My problem is in the end of the pictures, I put everything just to make easier to get what I'm saying)
My problem is here