Prefacing that everything stated here is only my own thoughts and opinions based upon my viewing of the anime eps 1-5. Nothing here is at all fact, even if stated as though it were. I could be totally wrong, I haven’t read the manga (no spoilers plz).
And this post contains spoilers for ep 1-5, ofc.
So! Why does Agott hate Coco? Well, she seems like an elitist who unfairly judges the girl for not being born into the magical world, and for skipping in line to become an apprentice without all the hard work Agott had to go through, right?
On the surface, it feels like a pretty shallow reason, but I think it’s actually more reasonable than it seems. Agott doesn’t just hate Coco because she’s ignorant of magic, but rather because she’s indifferent to the danger that improperly used magic, and forbidden magic especially, carries.
Now wait, that seems ridiculous, right? Coco destroyed her home and may well have killed her mother with forbidden magic. She knows better than anyone how dangerous it is!
That’s obvious to us, because we get to see it all play out from her perspective: her grief as Qifrey carries her away into the night, the way the scene loses all color when she gets anxious, how she cries alone in bed where no one can see…
But Agott doesn’t know any of that. From her perspective, Coco’s some random girl whose uncontrolled experiment essentially killed her own mother. She’s generally unexpressive/unreadable, and half-hearted about wanting to save her mom, yet simultaneously very excited about learning magic. Almost as if she were obsessed with magic, and without real concern for the harm it could cause. This is only further reinforced by her handling of the trial. Agott assumes Coco will see the spheres are beyond the reach of sylph shoes, stated to not be powerful enough to pass the trial in its current state, and just give up. But instead, Coco does something terrifying; she bets her own skin on a crazy half-baked contraption and a glyph she knows almost nothing about because getting to use magic matters more to her than even her own life does!
And that girl is living in their home. And she knows forbidden magic. It’s only a matter of time before they end up just like her mom.
Looking at it like that, it’s little wonder Agott wanted nothing to do with her.
So what gets her to finally come around to Coco? Well, the girl proves she’s clever and creative with magic, I’m sure that’s part of it. But more than that, I think it’s that Coco doesn’t want to hurt the dragon. A monster that is actively trying to kill them. And she doesn’t want to hurt it.
There’s no way someone like that could treat taking lives lightly. There’s no way she’s isn’t wracked with guilt about her mom, and there’s definitely no way she’d ever let it happen again. Agott herself wouldn’t have hesitated to blast that dragon with fire if it meant getting home safe. So Coco is taking magic safety even more seriously than even she is.
And she can’t hate someone like that.
Bonus note: Coco doesn’t get an “I’m sorry” from Agott, but at the end of ep 5 when Coco wakes up, she’s lying on top of three apprentice cloaks. All three of them would happily dirty the symbol of their witchhood if it’s for her, Agott included.
Bonus 2: I’m pretty sure Aggot doesn’t read non-verbal cues well. There are a ton of them, including some wonderful closeups on Coco’s hands, contextually hinting that she’s a lot more anxious and scared than she lets on. Tetia obviously picks up on them, given her horror when she realized Coco read her glance in ep 4. But I also think it’s part of the reason Tetia’s so much kinder to her throughout, despite her own initial distance after realizing Coco was the forbidden magic user.