In a manner characteristic of a coxcomb; in a foolishly vain or conceited way.
From 'coxcomical' + '-ly' (adverbial suffix). This is a standard adverbial formation from an adjective.
By the time you say 'coxcomically,' you've stacked three suffixes on the original 'coxcomb'—'coxcomb' + '-ical' + '-ly'—which shows how English builds longer, fancier words by layering meaning, and why these multi-suffix words sound especially formal and old-fashioned.
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