Under the influence of Morpheus? (6)

I believe the answer is:

drowsy

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'under' is the definition.
Both the answer and definition are adjectives. Perhaps there's a link between them I don't understand?

I don't understand how the remainder of the clue works.

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(Other definitions for drowsy that I've seen before include "Dull" , "Half asleep" , "Soporific" , "Comatose" , "Not being very alert" .)

I've seen this clue in the Evening Standard.

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