Angel learns what it really means to be a father, and most importantly, that even he can't be a father all by himself. The 249 years-old vampire has to face, for the first time, a small crying baby, and do it all by himself. From the moment he brings the baby to the hotel, he doesn't let go of him for a moment, and certainly doesn't let *anyone* even remotely touch his son, saying he promised his mother no one would touch that child and so it would be. His behavior verges obsessive-possessiveness and though in a sense, he can be understood, after everything he and Darla went through only or the baby to be born, it's still an impediment.
The first night, Angel doesn't close his eyes for a minute, later in the morning saying he spent the night watching his son. Just watching him. It shows he slowly starts to let go. He let the fears drift for that one night, or whatever was left of it, and spent these hours relishing his fatherhood, the miracle of his son…just watching him.
As soon as he is born, the infant draws to him every thinkable and unthinkable fiend. *Everyone* seem to want to possess that child and everyone will do *everything* to put their hands on him, from whatever reason they have. The Hyperion is no longer a safe harbor for the baby and for Angel, who will do everything to protect his son. And the crisis snaps him back to reality - like Cordelia said, he *can't* do this alone.
He and the gang come up with a plan, during which he'll keep busy whoever's after his son, and will lead them all into a trap, and while he does that, his friends will take the baby to the hospital for his first routine checkup. The plan runs smoothly, and Angel not only shakes the pursuers off of his and his son's backs, he also makes Linwood his son's Godfather (signing the "pack" by slashing his cheek with a knife as a payback for the baby's tiny scratch). From now on, if anything happens to the infant, the same thing will happen to him. That was enough to keep everyone possible away from the baby.
Afterwards, Angel arrives in the hospital, where he meets his friends with his son. He finally names him: Connor.
Cordelia: Get any sleep?
Angel: Nah, I was up all night watching him.
Cordelia: You know, if you'd let us take shifts we could...
Angel: No, I wasn't watching him like that, I was just...watching him. He...looks a little bit like me, don't you think?
Cordelia: Oh, yeah. Look at that brow. He's a miniature you. Why don't you hand him over for a while, catch some shut eye.
Angel: No. I'll sleep, when I know he's safe.
Cordelia: Which will be when? Around never? Angel, I understand you wanna protect him, but you gotta let go a little. Share part of the responsibility. You can't be everything for him.
Angel: I'm his only family. My job now is to be everything for him.
Cordelia: Really? Okay. Follow me. Come on.
Angel: Where are we going?
Cordelia: Come on.
Cordelia: We're going outside, where your son is gonna wanna go play. Where you have to rush him to the hospital if he gets sick in the daytime. But I see your point. You can't go outside during the day like other parents because you're a vampire. And even if you weren't you can't do everything for him.
Angel: If he has to get to the hospital at noon, on the sunniest day of the year, he'll get there. Even if I don't.
Cordelia: Angel, stop, Angel...