In the manner of a hawk, meaning watchfully, keenly, or with sharp attention.
Combining 'hawk' with the Old English suffix '-wise' meaning 'in the manner of' or 'direction of,' creating adverbs that describe behavior modeled after the referenced thing.
The suffix '-wise' was incredibly productive in English—people used to create adverbs by adding it to almost anything, so you'd see 'foxwise' (cunning), 'lambwise' (gentle), and 'hawkwise' (vigilant) used interchangeably!
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