The Cayley transform identifies the upper half-plane and unit disc by a Möbius map.Suggest a slogan
The fractional linear transformation \[ G(w)=i\,\frac{1-w}{1+w} \] maps the unit disc \(\DD\) biholomorphically onto the upper half-plane \(\HH=\ts{z\st\Im z>0}\). Its inverse is \[ F(z)=\frac{i-z}{i+z}, \] which maps \(\HH\) biholomorphically onto \(\DD\).
For \(w\in\DD\), \(w\neq-1\) and \[ \Im G(w)=\Re\frac{1-w}{1+w}=\frac{\Re\big((1-w)(1+\bar w)\big)}{\abs{1+w}^2}=\frac{1-\abs w^2}{\abs{1+w}^2}>0. \] For \(z\in\HH\), \(z\) is closer to \(i\) than to \(-i\), so \(\abs{F(z)}<1\). Solving \(z=i\frac{1-w}{1+w}\) gives \(z+zw=i-iw\), so \(w(z+i)=i-z\) and \(w=F(z)\); hence \(F\) and \(G\) are mutually inverse holomorphic maps.