Claiming children on taxes

Am I wrong?

Boyfriend of almost 10 years want to claim our 2 kids (8 and 5) on his taxes. I said no because his monthly expenses don’t change just because he has kids. He pays mortgage, car & insurance, phone, internet, lights, water and electric. None of these bills are because of kids. I pay for all child related expenses (daycare, medical, clothing, FOOD) AND we use my car whenever we go to appointments for the kids or anytime we go out of town.