Is $f \circ f^{-1}=id$ a sufficient criteria for diffeomorphism?
I recall having seen some version which had three functions in the composition.
But I also think of reading that $f \circ f^{-1}=id$ implies that both $f$ and $f^{-1}$ must be smooth. But I don't know if this also proves bijectivity?
Particularly, http://www.maths.adelaide.edu.au/michael.murray/dg_hons/node7.html
If $ f$ is a diffeomorphism $ f \circ f^{-1} = 1_{ \mathbb{R}^n}$
But is this a $\iff$ or $\implies$?
Also confused as to whether $f \circ f^{-1}$ really produces $\mathbb{R}$, since $f:U \rightarrow \mathbb{R}^n$ and $U \subset \mathbb{R}^n$. If Lemma 1.2 requires that $\phi^{-1}$ would be onto $\mathbb{R}$.
Invertibility is one requirement. Just as important is the requirement that both $f$ and $f^{-1}$ are $C^\infty$, and the definition of a function between manifolds being $C^\infty$ usually uses a certain composition of three functions.