Carrie’s in Julian’s good books because she’s confident, kind of hot-headed, and she gets points for making him laugh. She’s in the weirdly secretive camp along with Batwoman, and that’s a trait he’s starting to associate with pretty much anyone who has even the barest connection to Batman. It’s the thing he likes least about her, and particularly he doesn’t like not knowing her real name since she keeps her two identities very separate, and that means he only knows her in her costumed superhero form. He’s aware that’s not the real her. That doesn’t stop him from liking her, though. As may have been evidenced when he was injured, and actually told her the truth of what was wrong when he was dodging that question with every other person in that post, he trusts her judgement and trusts her not to betray a confidence. Probably because he knows she’s serious about keeping her secrets, so weirdly the thing he likes least about her also contributes to making him trust her. She isn’t someone he’d be likely to tell anything overly personal to, at least not right now, but she is someone he’ll be straight with. He’d also willingly fight alongside her, and he’d have her back if she needed him.