= Solution
Either equality condition says that the <holomorphic function> $h$ from part (i) has modulus one at an interior point: at zero in the <derivative> case, or at the given nonzero point in the other case. Since $|h|\leq1$ everywhere, the <maximum modulus principle> makes $h$ a constant $w$ with $|w|=1$. Therefore
$$
\boxed{f(z)=wz,\qquad |w|=1.}
$$
These maps are rotations of the <unit disk> and do achieve both equalities, so this also identifies all equality cases of the <Schwarz lemma>.
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