In a manner that is childish or resembling a baby; in an infantile or immature way.
From 'babish' (adjective) plus the '-ly' suffix (Old English origin, from 'like' or 'resembling'), which standardly converts adjectives to adverbs describing how something is done.
The '-ly' adverb suffix is so common that we barely notice it, but it's one of the most useful tools in English—it lets us turn any adjective into a description of how an action happens, making our language incredibly flexible.
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