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Related to cozy up: cosy up
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1. To move close (to someone) in order to snuggle or simply be close. I left the room when my sister and her boyfriend started cozying up. The puppies cozied up to their mother.
2. By extension, to become friendly with someone, usually in an attempt to obtain something. Since I'm deciding who gets the promotion, Crystal has been cozying up to me by buying me coffee all week.
cozy up(to someone)
1. Lit. to snuggle up to someone as if to get warm. The children cozied up to their mother. They cozied up to each other.
2. Fig. to try to get in good with someone; to try to increase one's influence with someone by being extra nice and friendly. The salesman tried to cozy up to the customer.
Try to get on friendly or intimate terms, ingratiate oneself. For example, That new woman is always cozying up to one or another club member so she'll be asked to join . [Mid-1900s]