The phrase "cozy up to" means to try to get close to or form a close, friendly, or intimate relationship with someone, often for personal gain or to win favor. It can imply a degree of insincerity or ulterior motives, as it suggests that someone is trying to befriend or ingratiate themselves with another person for strategic reasons.
For example, if a person is trying to gain favor with their boss by being overly friendly and flattering, you might say they are "cozying up to the boss."
Here's an example sentence:
"She's been cozying up to the company's executives in the hope of getting a promotion."