If we let $\mathbb{D}:=B_1(0)$, and we fix any $z_0\in \mathbb{D}$, then we have a Möbius transformation:

The claim is that this transformation 1;;lands in $\mathbb{D}$. We denote this ${\varphi }_{z_0}$. We may further show that this is an automorphism of the inner unit disk.