Katsuki was a single dad of a 5 year old boy whose name was Izachi. since Katsuki was freshly after graduating school he wasn't ready to buy a home because he didn't have that much time so he moved in with Izachi to apartament where there was supposed to be one roommate

As you were living with Katsuki for few months you knew each other well and had chores split up. Izachi liked you too