For \(a\in\DD\), the Blaschke factor, or hyperbolic translation, at \(a\) is \[ \psi_a(z)\coloneqq\frac{a-z}{1-\bar a z},\qquad z\in\CC,\ \bar a z\neq1. \] For \(n\ge1\), points \(a_1,\ldots,a_n\in\DD\), and \(\lambda\in\CC\) with \(\abs{\lambda}=1\), the finite Blaschke product with zeros \(a_1,\ldots,a_n\) is \[ B(z)\coloneqq\lambda\prod_{k=1}^{n}\psi_{a_k}(z)=\lambda\prod_{k=1}^{n}\frac{a_k-z}{1-\bar a_k z}. \]
Some authors take the Blaschke factor to be \(\phi_a(z)\coloneqq\frac{z-a}{1-\bar a z}=-\psi_a(z)\); the two conventions differ by the unimodular constant \(-1\), which is absorbed into \(\lambda\) in a Blaschke product.
Let \(a\in\DD\).
For \(\abs{z}\le1\), \(1-\abs{\psi_a(z)}^2=\frac{(1-\abs{a}^2)(1-\abs{z}^2)}{\abs{1-\bar a z}^2}\); hence \(\psi_a(\DD)\subseteq\DD\) and \(\psi_a(S^1)\subseteq S^1\).
\(\psi_a\circ\psi_a=\operatorname{id}\) on \(\overline{\DD}\), so \(\psi_a\) is a biholomorphism \(\DD\to\DD\) exchanging \(0\) and \(a\).
\(\psi_a\) preserves the hyperbolic metric \(\frac{\abs{dz}}{1-\abs{z}^2}\) on \(\DD\): \(\frac{\abs{\psi_a'(z)}}{1-\abs{\psi_a(z)}^2}=\frac{1}{1-\abs{z}^2}\) for \(z\in\DD\).
Every finite Blaschke product \(B\) is holomorphic on a neighborhood of \(\overline{\DD}\), maps \(\DD\) into \(\DD\) and \(S^1\) into \(S^1\), and its zeros in \(\DD\) are exactly \(a_1,\ldots,a_n\).
Since \(\abs{a}<1\) and \(\abs{z}\le1\), \(\abs{\bar a z}<1\), so \(\psi_a\) is holomorphic on the disc \(\abs{z}<1/\abs{a}\), which contains \(\overline{\DD}\).
Expanding, \(\abs{1-\bar a z}^2-\abs{a-z}^2=1+\abs{a}^2\abs{z}^2-\abs{a}^2-\abs{z}^2=(1-\abs{a}^2)(1-\abs{z}^2)\); divide by \(\abs{1-\bar a z}^2\). The right side is positive for \(\abs{z}<1\) and zero for \(\abs{z}=1\).
For \(\abs{z}\le1\), \[ \psi_a(\psi_a(z))=\frac{a(1-\bar a z)-(a-z)}{(1-\bar a z)-\bar a(a-z)}=\frac{z(1-\abs{a}^2)}{1-\abs{a}^2}=z. \] With (i), \(\psi_a\colon\DD\to\DD\) is holomorphic with holomorphic inverse \(\psi_a\), and \(\psi_a(0)=a\), \(\psi_a(a)=0\).
By the quotient rule, \(\psi_a'(z)=-\frac{1-\abs{a}^2}{(1-\bar a z)^2}\); divide by the expression for \(1-\abs{\psi_a(z)}^2\) in (i).
Each factor is holomorphic near \(\overline{\DD}\), has modulus less than \(1\) on \(\DD\) and equal to \(1\) on \(S^1\) by (i), and vanishes on \(\overline{\DD}\) only at its own \(a_k\); multiply, using \(\abs{\lambda}=1\).