It's all fun and memes to joke about how Caitlyn immediately switched sides the moment she hears Vi call her cupcake again. And I do it too. But to me, the switch wasn't abrupt at all.
From the first scene in episode four, Caitlyn already says that the martial law has been going on much longer than she wanted. In the next scene, you see her standing up to Ambessa, using her words back at her, for which Ambessa calls her fearless. It shows that Caitlin has not given up control and is not being puppeteered by Ambessa. She's in control still, but her singular focus is still on Jinx like it was in act one.
Vi was right that Caitlyn is behaving like Jinx, but it was probably the worst thing Vi could have said to her, no matter how true, cause Caitlyn was too deep in her quest for 'justice' in her grief. That's why she reacted so violently.
Jinx is the target of her obsession, or her in her mind for justice, even though what she really wants is vengeance, and being compared to Jinx cut deep. Vi had noticed it before in season 1:
In the beginning of episode four, she feels that it's been going on too long, but even after all those months her obsession is still stronger. Her obsession with getting vengeance, finding Jinx, killing Jinx, is still stronger than her moral compass, which she has abandoned to avenge her mother and fill her mother's shoes.
But over the course of the second act before she sees Vi again, there are multiple things that happen that make Caitlyn understand she has to stop, that what she's doing isn't right or justifiable.
Some of these were not initially clear to me when I first watched it. but one thing I realized today while I was working out is that one crucial moment for Caitlyn's arc in act two is with Singed.
Taking her singular focus off Jinx for a moment, Caitlyn does what she does best and figures out who Singed is, and what he's responsible for. She uncovers that Singed is actually the formerly revered alchemist Dr. Corin Reveck, who used to work with Heimerdinger at the academy. Arcane implies he is from Piltover and became a Zaunite after being kicked out of Piltover. He helped build up Piltover to what it is, and now he is ravaging Zaun with shimmer, weapons and monsters.
When she calls him a monster and asks him "why do all this?", he tells her he did it 'for love'. Out of love for his deceased daughter.
And I think that that is one of the crucial moments where Caitlyn starts to realize that what she is doing is not very different from him.
She's doing things her old self would never have approved, of out of love and grief for her deceased mother.
When she sees Vi again, she doesn't suddenly switch sides, she's already been doubting her chosen course of action. She initially says "you can't be here", because Caitlyn is trying to believe she's doing the right thing still, and knows Vi is a threat to that belief/lie.
Vi calling her cupcake though, is one of the last drops in the bucket. She left Vi to do what she thought she needed to do, what her mother might have done. What she's doing out of guilt for not saving her mother when she had the chance. She even tried to be with a girl her mother would have approved of.
When Vi calls her cupcake, all the feelings she's been trying to stuff away come flooding back. She snaps out of this self-imposed love-deprived state she's been in, and she can no longer deny she loves Vi.
When Vi tells her that she's trying to save her dad, I think that is the moment she actually switches sides. That was the last drop. That is the moment where she realizes there are still people she loves, who love her, who are still alive and they need her help, and that is more important than her horrible campaign for vengeance out of grief, otherwise she's no different than Singed.
