  Let $\varphi:\mathbb{D} \to \mathbb{D}$ be a holomorphic map with a fixed
point $\alpha\in\mathbb{D}$ such that $0\leq |\varphi'(\alpha)|<1$. We show
that the spectrum of the composition operator $C_\varphi$ on the Fr\'echet
space $ \textrm{Hol}(\mathbb{D})$ is $\{0\}\cup \{
\varphi'(\alpha)^n:n=0,1,\cdots\}$ and its essential spectrum is reduced to
$\{0\}$. This contrasts the situation where a restriction of $C_\varphi$ to
Banach spaces such as $H^2(\mathbb{D})$ is considered. Our proofs are based on
explicit formulae for the spectral projections associated with the point
spectrum found by Koenigs. Finally, as a byproduct, we obtain information on
the spectrum for bounded composition operators induced by a Schr\"oder symbol
on arbitrary Banach spaces of holomorphic functions.
