We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity. We have shown in question (19) that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$. Conclude that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$.