To fan or cool with a fan; to provide with a fan or fanning motion.
From the prefix 'be-' (Old English, indicating provision or action) combined with 'fan' (from Latin 'vannus,' a winnowing fan). The formation parallels other causative 'be-' verbs that describe providing or using tools.
Befan is a perfectly logical word that simply isn't used anymore—in an earlier English, you could befan someone in hot weather just as you might befriend them today, showing how productive 'be-' was before it fossilized into only a few expressions.
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