Here is my attempted solution: Let $f$ be a fractional linear transformation from $\mathbb D$ onto $\mathbb D$. Then $f$ is analytic. Consider, $g= \phi_{f(0)} o f$. Then $g$ maps $\mathbb D$ onto $\mathbb D$. By Schwartz lemma, $|g^{-}(z)| \leq |z $ for all $z \in \mathbb D$. So, $|g(z)|=|z|$ for all $z \in \mathbb D$. Then, by 2nd part of Schwartz lemma, $g(z)=c_{1}z$ such that $|c_{1}=1|$ for all $z \in \mathbb D$. So, \begin{align} f(z) &=\phi_{f(0)}^{-1}(c_{1}z)\\ &= \frac{c_{1}z+\alpha_{1}}{1+ \alpha_{1}c_{1}z} \\ &=c_{1} \frac{z+ \bar{c_{1}} \alpha_{1}}{1+ \alpha_{1}c_{1}z} \end{align}
Here $\alpha_{1}=f(0)$ and $|\bar{c_{1}}|=1$ Is my approach correct?