= Solution
Existence follows from the three-transitivity of <Möbius transformations>: one can send three distinct source points to $0,1,\infty$ and then send those to the three distinct target points.
For uniqueness, suppose $S$ and $T$ have the same values at three distinct points. Then $M=T^{-1}S$ fixes those points. If
$$
M(z)=\frac{az+b}{cz+d},
$$
the finite fixed points satisfy $cz^2+(d-a)z-b=0$. A nonidentity Möbius map therefore has at most two fixed points on the Riemann sphere. Since $M$ has three, it is the identity, and $S=T$.
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