OHHHH-I kinda was vibing with the idea that they know they're each other's soulmates and are in a relationship, but Katsuki treats him like shit/ignores him when they're in school and pretends they aren't in a relationship.
He'd been allowed into U.A.-on one condition. The shock collar was a special invention from Research and Development-it was wired to go off every time he lost control of his powers. just small shocks at first to keep him in check, then bigger ones if he couldn't keep himself under control. But it could really, REALLY hurt if he got out of control.