In a manner that expresses disapproval, apology, or humble protest; in a self-deprecating way.
From deprecating (adjective) + -ly (adverbial suffix). The adverbial form allows the meaning to modify verbs and show how an action is performed.
The British use 'deprecatingly' to describe a very specific social behavior—that polite, apologetic way of disagreeing that avoids confrontation, a cultural pattern reflected in the language.
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