c. 1200, "willing to serve (someone); willing to fulfill an obligation," from Old French obedient "obedient" (11c.), from Latin oboedientem (nominative oboediens) "obedient, compliant," present participle of oboedire "to obey" (see obey).
common adverbial suffix, forming from adjectives adverbs signifying "in a manner denoted by" the adjective, Middle English, from Old English -lice, from Proto-Germanic *-liko- (cognates: Old Frisian -like, Old Saxon -liko, Dutch -lijk, Old High German -licho, German -lich, Old Norse -liga, Gothic -leiko); see -ly (1). Cognate with lich, and identical with like (adj.).
Weekley notes as "curious" that Germanic uses a word essentially meaning "body" for the adverbial formation, while Romanic uses one meaning "mind" (as in French constamment from Latin constanti mente). The modern English form emerged in late Middle English, probably from influence of Old Norse -liga.
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Definitions of obediently from WordNet
in an obedient manner;
obediently she slipped off her right shoe and stocking