Maybe there is a competition or a challenge that only Kakashi can help Naruto with. Naruto is always eager to improve his skills and prove himself. If Kakashi has some special knowledge or training method related to this challenge, Naruto would definitely go after him to get that edge.
One reason could be for guidance. Naruto often looks up to Kakashi and might need his advice on a difficult situation, like dealing with a new powerful enemy.
In fanfiction, Naruto could be driven to the library by his desire to understand the world outside of Konoha better. He has seen so much in his adventures, but there's still a lot he doesn't know. The library could have books about different lands, their cultures, and their forms of ninjutsu. By studying these, Naruto can become a more well - rounded ninja and gain a broader perspective on the ninja world as a whole.
It's possible that there's a strange illness going around the village and Naruto goes to the hospital to see if he can offer any help. He has a strong sense of justice and would want to do whatever he can to stop the spread of the illness and make sure everyone gets better.
A 'Naruto modeled after Kakashi' fanfiction could see Naruto with a changed appearance to some extent. He might start wearing his headband in a similar way to Kakashi, or even have a similar style of clothing. In terms of his ninja skills, he could master the Sharingan - like abilities in some form, perhaps a new jutsu inspired by it. His mentoring style could also change, becoming more like Kakashi's, being tough but fair with his students or comrades.
In some fanfiction concepts, Naruto may go to Orochimaru for the sake of a friend. For instance, if a friend is dying and Orochimaru has the cure or a way to save that friend. Naruto is very loyal and would do almost anything for his friends. So, he might take the dangerous step of going to Orochimaru, even though he knows how dangerous and untrustworthy Orochimaru is.
Kakashi might be initially surprised. But then, he would probably use it as a teaching moment for Naruto. He could say something like 'Naruto, stealing is not the way of a ninja.' And then give him a punishment like extra training.
Well, it could be that Kakashi, being a character who values the integrity of the original Naruto narrative, sees fanfiction as a deviation. Fanfiction might include out - of - character scenarios for him or the other characters. He might also dislike the fact that some fanfictions can be of very poor quality in terms of writing and plot, which could be a reason for his dislike.
Kakashi would likely be very concerned. He would immediately assess Naruto's injuries. If they were serious, he would use his medical skills to start treating him right away. His first reaction would be to make sure Naruto was stable and out of immediate danger.
Naruto might be the better singer in terms of energy and enthusiasm. His loud and boisterous nature could translate into a powerful and engaging singing voice. He'd probably belt out songs with great passion, getting the audience pumped up.
Their relationship would likely become very close. Naruto would probably start to see Kakashi as a father figure. Kakashi, in turn, would become more protective of Naruto.
Maybe out of curiosity. They want to see how the fans view Kakashi, who was an important part of their lives. Kakashi was Minato's student and Kushina knew him well too. So they are interested in the different perspectives presented in the fanfiction.