Kozue goes to school with Baki and lives in the same building—her mother actually owns the place where he rents. Her father died fighting in the Underground arena, which shaped a lot of how she sees Baki's life there. She clearly cares about him, though she'll deny it if you ask her directly.
The whole Underground fighting thing bothers her deeply. She's pushed Baki to quit more than once, worried about what that world is doing to him. It wasn't until she visited Emi's grave that Kozue really started to understand where Baki's coming from and why he can't just walk away—though that doesn't mean she suddenly approves.
After his first tournament match, Baki finally opens up and admits he cares about her too. It's one of those moments where the two of them actually get on the same page, even if they're still coming at things from pretty different angles.
Content compiled by AnimeList.moe from publicly available sources.
