In some fanfictions, Aizawa might be under the influence of a villain's quirk that makes him act against his will. So, he kidnaps Izuku, but later on, he realizes what he's done and tries to make amends. This could lead to a story full of drama and redemption.
Perhaps Aizawa kidnaps Izuku because he discovers a secret about Izuku's origin that is related to a greater threat. He wants to keep Izuku away from the public eye until he can figure out how to deal with this new information. This could involve a lot of mystery and suspense as Izuku tries to figure out why his teacher has done this.
It could be Aizawa himself if he's being controlled by something or has some sort of ulterior motive that is not in Izuku's best interest. But if not Aizawa, then there could be a hidden villain who is manipulating Aizawa to kidnap Izuku.
Well, it could be about Izuku confronting Aizawa over some misrepresentation in the fanfiction. Maybe Izuku doesn't like how Aizawa is being portrayed as too strict or unfair in the fic.
One scenario could be that Izuku is on the roof to get some fresh air and clear his head. Aizawa, being the observant teacher he is, follows him there. He could then share some of his own experiences with Izuku, like how he overcame difficulties in his youth, which would inspire Izuku and strengthen their relationship in the fanfiction world.
One reason could be that Izuku has been pushed to his limit. Maybe Aizawa has been overly critical or strict in a way that Izuku can't take anymore. For example, if Aizawa constantly doubts Izuku's abilities despite his efforts, it could lead to Izuku snapping.
If All For One kidnaps Izuku in fanfiction, Izuku's friends would surely try to rescue him. They could form a plan, with Bakugo being especially determined as he has a long - standing relationship with Izuku. All Might might also get involved, using his remaining power to try and save his successor.
One possible plot could be that due to some false accusations of improper teaching methods, Aizawa gets fired. His students then band together to prove his innocence. They collect evidence of his effective teaching, like how he has helped them improve their quirks and grow as heroes in training. Eventually, they manage to clear his name and he is reinstated.
One good one might be 'Aizawa's Guidance for Izuku' which focuses on their unique mentor - mentee relationship evolving in an interesting way. It has great character development.
Perhaps Izuku was skipping class and went to the roof. Aizawa noticed his absence and suspected he might be there.
Aizawa may also interact with Izuku's family during family events or gatherings in the fanfic. He could be a guest and during that time, he could share his insights on Izuku's growth and development. This would not only strengthen his relationship with Izuku but also with his family as a whole.
Aizawa is known for his strict and somewhat reserved nature. This would have a significant impact on Izuku. Izuku, who is often eager and a bit impulsive, would learn to be more patient under Aizawa's guidance. Aizawa's ability to see the potential in his students despite their flaws would also make Izuku more confident in his own abilities. He would know that his father figure believes in him, and that would encourage him to keep pushing himself. Moreover, Aizawa's dry sense of humor might also influence Izuku's own sense of humor, making him a bit more sarcastic in a light - hearted way.