  Let $\varphi$ be a holomorphic map which is a symbol of a bounded composition
operator $C_\varphi$ acting on the Hardy-Hilbert space of Dirichlet series. We
find a K\"onigs map for $\varphi$. We then deduce several applications on
$C_\varphi$ (e.g. on its spectrum, on its dynamical properties). In particular,
we study for a large class of symbols $\varphi$ if the associated composition
operator has a minimal commutant.
