督

Bottom 目 is the indexing eye radical — a vertical eye-frame with horizontal lines for the iris and pupil. Marks 督 as an act of watching: supervision is fundamentally vision-with-authority. Compound logic: under-the-eye = observed, supervised. Same radical family: 看 look, 眼 eye, 睡 sleep, 瞎 blind, 盯 stare.

Top 叔 supplies the sound — shū drifting to dū through Old Chinese velar-cluster split, the same shift seen in 督促 dūcù urge. Pure phonetic loan; the 'uncle' meaning of 叔 contributes nothing. The shū/dū correspondence reflects an Old Chinese initial that resolved differently across modern Mandarin readings of related chars.