曙 = 日 (sun) + 署 (signpost / arranged in order). Picture the sun stepping into its appointed place at the horizon, daybreak.
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曙 is bound, you almost never see it alone in modern Chinese. It lives in compounds: 曙光 (shǔguāng, 'first light / hope'), 曙色 (shǔsè, 'dawn light'), 拂曙 (fúshǔ, 'just before dawn').
Left sun radical, pictograph of the sun with a dot marking its substance. The indexing radical, putting 曙 in the daylight family with 明 bright, 昏 dusk, 晨 dawn, 晓 daybreak. 曙 specifically means "first light of dawn", the moment the sun just clears the horizon.
Right 署 supplies the sound, shǔ carried over without change. 署 itself means "government office" and has no semantic role in 曙. Same phonetic frame appears in 薯 yam, 暑 hot season. Pair 曙 with 暑 for memorisation: both have sun as radical and 署-style phonetic.
