Here are some example sentences with cosy up:
After a long day at work, all Sarah wanted to do was cosy up on the couch with a good book and a warm blanket.
The newly adopted kitten cosied up to its owner, purring softly as it snuggled against her.
As the temperature dropped, the family gathered around the fireplace to cosy up and enjoy some hot chocolate together.
In an effort to secure a promotion, Tom tried to cosy up to his boss by praising her ideas and offering to help with her projects.
You can see that it is cosy up to which has the meaning of getting some benefit, and even then, it is not always the case.