In the manner or style of a college or collegium; at the level or in the context of collegiate education.
From collegiate (adjective meaning of or relating to a college) + -ly. Collegiate comes from Latin collegium plus -ate, then -ly converts it to an adverb describing how something is done in a college-like way.
You might see 'collegiately' used in academic writing about sports—'competing collegiately means representing your university'—but it's getting rarer as plain English prefers 'at the college level' instead.
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