Haakaa milk question

I’m 9 days pp, I had to start pumping bc my little one went from 6lbs 10oz to 6lbs 1oz. I told her dr I felt my milk was in and i really didn’t want to pump bc I had a really bad oversupply with my first kiddo. But I pumped anyway to assure my milk was in (and it is.. I already get 5oz when I pump). My baby will only drink off 1 breast so I’ve been using the Haakaa on the other to help with engorgement when she’s nursing. My question- does the Haakaa only pull the foremilk? I assume you can’t just freeze the Haakaa milk alone, you would need to mix it with pumped milk to get the fatter milk that comes later? I never used the Haakaa with my first baby.