Loki on Disney Plus is the perfect vehicle to explore a character who's been in the MCU for 10 years, but has never had the spotlight pointed directly at him before. Loki has been a fan favorite for years, and has had a great character arc over the course of the first three phases of the MCU, but there was never really time to dig very deep into his character until now. The show displays a Loki who is much more nuanced than we've seen in the past, and who is finally able to open himself up to other people.
Some fans have claimed that Loki feels out of character in the show, especially considering the fact that this series is supposed to take place directly after The Avengers. The way he acts in that film is completely different than how he acts in Loki, and just in general he seems so much different than how we've seen him in the MCU before, to the point where a portion of the fanbase has complained that this is a problem with inconsistent writing. However, if you take a deeper look at Loki's character through the MCU, and consider what he goes through in the show, it makes total sense that he would act the way he does in Loki, and if anything, he's more in character than ever before.
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Avengers Loki is the most extreme version of the character that appears in the MCU. He seems a little bloodthirsty and crazed, which is a far cry from the usual measured approach and relative calmness that he displays in his other MCU appearances. In those, he only really goes to extreme emotions when he's deeply upset. The Avengers is a low point for the character, after having his world rocked and feeling like his family rejected him, so it makes sense that this would be the most extreme version of the character. On top of this, Marvel has confirmed that he was being influenced by the Mind Stone (which was in his scepter) during the events of this movie. He wasn't being controlled, per se, but the Mind Stone heightens feelings that are already present, so his rage and resentment were only being magnified by the Stone, resulting in the Loki that we see in that movie.
So sure, it might seem like a huge personality shift to go from Avengers Loki to show Loki, because he's much more adjusted and just generally a better person, but it makes sense that once he's away from the influence of the Mind Stone, he has more of his old personality back. There's also the fact that Mobius shows him the highlight (or perhaps lowlight) reel of his life, and makes Loki aware of the fact that his actions only ever hurt people and eventually even got him killed. The character development that happens in that first episode might seem rapid, but it makes sense that after being faced with your fate like that, you might choose to start being a better person in hopes that you can avoid that end.
In fact, Loki acts more in character (in comparison to his other MCU appearances) in the show than he does in The Avengers, so it's unfair to expect that his personality in Loki should match his most extreme of appearances. In the Thor movies, Loki has always been quite level. Sure, he thrives on a bit of chaos and is playful (especially in the latter two Thor films), but he has this sort of softness to him, and you can see that he's a character who has some deep emotional wounds. The way he acts in the show sort of feels like a cross between Loki in the first Thor movie (who is sensitive and emotional) and Ragnarok Loki (who is more unpredictable and silly).
Even in Infinity War, we see a version of Loki who has connected to his brother again, and who is willing to give his life for a good cause. Sure, he had a few movies of character development while show Loki only had hours to get to a similar point, but both of these appearances feel more similar to each other (and are more representative of Loki as a character) than his run in Avengers ever did.
Some of the criticism lobbied towards the writing of the character revolves around his choice to try and stop Sylvie from killing He Who Remains in the final episode of Loki. Some fans believe it's out of character for him to side with the TVA, and with law and order in this moment, rather than letting the multiverse have free rein. True, that option does seem more in line with his chaotic side, but Loki's character has always been focused on self-preservation. He knows that unleashing the multiverse will release all these variants of He Who Remains into the world, and that a multiversal war will just cause a bigger problem and more death and destruction. He states in the first episode that he doesn't want to hurt people, and he can see the bigger picture of what will happen if Sylvie kills the one person who's keeping the multiverse in check.
This also tracks with his character because he does something similar in the first Thor movie. When Thor wants to start a war with the frost giants, it's Loki who tries to dissuade him. Loki never wanted to be involved in a war, and in The Avengers, he was likely convinced that war would be a good idea (both by outside forces and the influence of the Mind Stone), and it was more about him taking control than anything. In fact, show Loki - who is immediately post-Avengers - knows exactly where violence and war get a person, and he doesn't want to end up there again, and he doesn't want Sylvie to take that same path. In this way, it makes total sense that he would oppose her on that front.
While there's certainly moments of character development from Loki to criticize, it seems very reductive to say that Loki is out of character in the show just because he's not acting like Avengers Loki. If anything, that's the entire point of the show, to remind the viewer that Loki is so much more than the conniving villain we saw him as in The Avengers. His character development may have come somewhat rapidly, but it makes sense in the context of Loki, and it doesn't waste time spending the whole show developing a character that we already saw developed once in the MCU. They get that out of the way in the beginning so that they can use the show to go even deeper with the character than we got to see with Sacred Timeline Loki, and give him the development that he's deserved for 10 years.