DadyCoool asked: I wonder how much canon!Azula thought about Zuko during his exile. How long did it take for the "I'm not being dragged down anymore" feeling to fade, and how long did it take for her to stop expecting to see him behind every corner? Did she turn to point something out to him, only to be disappointed immediately?
On a related note, how much effort did she put into his capture when we first met her? Did she delay it long enough for him to escape? She did spend a lot of time with him after the Catacomb Fight, as if she was clinging to him in her own way.
Azula definitively had these moments in which she just forgot her brother was no longer around, but she didn't think much of it at first since she had already expected that (it probably happened when Lu Ten died, when her mother disappeared, when her friends left). But she starts noticing that she is more bored, more irritable and lonelier since Zuko's banishment, and even though she doesn't want to admit it, deep down she knows that she misses him. She hates it and will definitively not let it show since her father would likely see it as weakness or even treason to care about someone he decided to banish, but she can't help but be at least a little bitter. Her dum-dum was a loser, whinny and pathetic, but he was fun to be around and be fun to tease.
Because of it, she had absolutely no problem trying to capture him. He would be a prisoner and "no longer" her equal (he was never truly her equal, nobody was) and he would obviously be more furious at her than he had ever been their whole lives, but at least he would be home. She would get to see him. If she wanted to talk to him, tease him, irritate him, torment him, give him advice, or do literally anything to him, she would be able to. He'd be her little pet or her toy.
But as he keeps trying to fight her, despite him always losing, Azula can't help but to respect him a little more. He went from a boy who was "too weak" to fight his father to save himself in their Agni Kai, to fighting every man in her ship to get to her and make her pay for lying to him, tracking and chasing both her and the Avatar and facing them both at the same time (and actually holding his own), and then when she tricked him into going to the earth king's palace, he openly challenged her to an Agni Kai and tried to face both her and the Dai Li at the same time instead of escaping with their uncle. Was it stupid? Yes. But it was brave and he was somewhat competent in his own way. He has potential and she recognized that. Which is why she gave him a chance to "redeem" himself - a chance that he took, helping her out when she needed him most.
All of this combined explains why she was much kinder to him than she had been when he was banished. Even the whole telling Ozai that Zuko was the one to kill the Avatar only happened after they were home and Zuko lied to her about there being no chance of Aang having survived - she might care for him, love him even, but she won't just let him throw her under the bus, even if he likely didn't realize what he was doing. That incident aside, she was constantly trying to reach out to him. Considering he couldn't help but care when she was falling "to her death" and then needed emotional support when he saw her cry after their Agni Kai, I say she managed to make it just as impossible for Zuko to shake away his feelings for her as it is for her shake away her feelings for him.