Let \(z_1, z_2, z_3\in\CC\) be distinct. The Möbius transformation \[ T(z) \coloneqq { (z-z_1) (z_2-z_3) \over (z-z_3) (z_2 - z_1)} \] satisfies \(T(z_1)=0\), \(T(z_2)=1\), and \(T(z_3)=\infty\), and it is the unique Möbius transformation with these values. It is denoted \((z; z_1, z_2, z_3)\).
Consequently, for distinct \(w_1,w_2,w_3\in\CC\), the map \[ S\inverseof{\coloneqq(\,\cdot\,; w_1, w_2, w_3)} \circ (\,\cdot\,; z_1,z_2, z_3) \] is the unique Möbius transformation with \(S(z_j)=w_j\) for \(j=1,2,3\).
Substituting \(z_1\), \(z_2\), \(z_3\) into \(T\) gives \(0\), \(1\), \(\infty\). If \(T'\) is another Möbius transformation with the same values, then \(R\coloneqq T'\circ \inverseof{T}\) fixes \(0\), \(1\), and \(\infty\). Fixing \(\infty\) forces \(R(z)=az+b\); fixing \(0\) forces \(b=0\); fixing \(1\) forces \(a=1\); so \(R=\id\) and \(T'=T\). The map \(S\) is a composite of Möbius transformations, and it sends \(z_1,z_2,z_3\) to \(0,1,\infty\) and then to \(w_1,w_2,w_3\). If \(S'\) is another Möbius transformation with \(S'(z_j)=w_j\) for \(j=1,2,3\), then \((\,\cdot\,;w_1,w_2,w_3)\circ S'\inverseof{\circ(\,\cdot\,;z_1,z_2,z_3)}\) fixes \(0\), \(1\), and \(\infty\), so it is the identity by the same argument, and \(S'=S\).
In [@Ahl79, Chapter 3, Section 3.2, Definition 12], the cross ratio \((z_1,z_2,z_3,z_4)\) has four arguments: it is the image of \(z_1\) under the Möbius transformation carrying \(z_2, z_3, z_4\) to \(1, 0, \infty\). In that notation, \((z;z_1,z_2,z_3)=(z,z_2,z_1,z_3)\).