1.VerbTo come physically close to, using body language in an attempt to persuade another (often hesitant) party to snuggle or embrace.
"He forgot the whole argument when she cosied up to him on the couch.; The kitten cosied up to the gruff old hound, and all resistance disappeared."
2.VerbTo form a relationship with some one or some thing for the purpose of obtaining some benefit.
"It is often suggested that politicians are too inclined to cosy up to big business in order to receive funds for election expenses.; The sales guys might be able to cosy up to the customers, but they are inexperienced in technical support."