set cap for
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set (one's) cap for (someone)
dated To try to attract, secure, or win someone as a romantic partner or spouse. (Said especially (though not exclusively) of a woman in pursuit of a bachelor.) Well, if he insists on remaining so inhospitable, then I shall simply set my cap for a man with a greater sense of charm and decency. The ladies of this town shall all be setting their caps for Mr. Rutherford, now that his inheritance has left him quite wealthy.
set (one's) cap for
To attempt to attract and win as a mate.