Biholomorphisms of the polydisk

Let $\mathbb{D}$ denote the unit disk in the complex plane, equipped with the Poincare metric. Let us denote the group of biholomorphisms of $\mathbb{D}$ by $Aut(\mathbb{D})$.

Suppose $F: \mathbb{D}^n \rightarrow \mathbb{D}^n$ is a biholomorphic map. Then it is a standard fact that $F$ must be the composition of $n$ biholomorphisms of the disk $\phi_1,\ldots,\phi_n \in Aut(\mathbb{D})$ and an element of the symmetric group $\sigma\in\mathcal{S}^n$:

$F(z_1,\ldots,z_n) = (\phi_1(z_{\sigma(1)}),\ldots, \phi_n(z_{\sigma(n)}))$

and further that the group of biholomorphisms of $\mathbb{D}^n$, $Aut(\mathbb{D}^n)$ is the semidirect product of $Aut(\mathbb{D})^n$ and $\mathcal{S}^n$.

BUT, while this seems completely plausible to me, I can't quite convince myself that it is true, nor can I find a satisfactory reference. Any suggestions?

via Recent Questions - Mathematics - Stack Exchange http://math.stackexchange.com/questions/289669/biholomorphisms-of-the-polydisk