Example about hyperbolicity. $\def\abs#1{\left|#1\right|}$I would like to understand this example:


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*Why is the following set a hyperbolic manifold?


$X=\{[1:z:w]\in \mathbb{CP}_2\mid0<\abs z< 1, \abs w  < \abs{\exp(1/z)}\}$
It's an examples given in the book Hyperbolic Manifolds and Holomorphic Mappings: An Introduction by Kobayashi, in order to give a counterexample of an optimistic generalization of the Big Picard Theorem. They claim that it is biholomorphic to $\mathbb{D}\times\mathbb{D}^*$. I dont understand why. 

$\mathbb{CP}^2$ is a natural complex manifold where the chart are given by the maps :
$\begin{array}{lclc} \varphi_i:  & U_i:=\{[z_0:z_1:z_2]\in \mathbb{CP}^2 \ | \ z_i\neq 0\} & \longrightarrow & \mathbb{C}^2 \\  & {[z_0:z_1:z_2]} & \longmapsto & (\dfrac{z_j}{z_0},\dfrac{z_k}{z_0}) \end{array}$
where $j,k\neq i$.
However, you can write $U_1$ as $\{[1:z:w]\in \mathbb{CP}^2\}$ and by definition of a biholomorphic map between two complex manifolds, it tells you that $X$ is biholomorphic to $\varphi_1(X)$. 
The map $(z,w)\in \varphi_1(X)\mapsto (z,we^{-\frac{1}{z}})\in \mathbb D^\star\times \mathbb D$ is clearly a biholomorphism.
So $X$ is biholomorphic to $\mathbb D^\star\times \mathbb D$.
Since $\mathbb D$ and $\mathbb D^\star$ are hyperbolic manifolds then so is $\varphi_1(X)$ and consequently $X$ is a hyperbolic manifold.
