He may be surprised. Fanfictions can have some really out - there ideas that he might not have considered before. For instance, a fanfiction might have Naruto suddenly having a new, super - powerful jutsu that was never seen in the original series.
Well, one similarity could be their determination. In the fanfic, like in the original series, both Naruto and Kakashi are very determined. Naruto is always striving to become Hokage and Kakashi has his own goals and missions he's dedicated to. They also might share similar combat styles in the fanfic, perhaps Naruto starts to adopt some of Kakashi's strategic thinking during battles.
Well, Kakashi might be really engrossed. Since Jiraya is a great writer in the Naruto world, Kakashi could be learning a lot from those novels. Maybe he finds some new ninjutsu concepts or interesting stories about the shinobi world in them.
Sasuke could be a popular character. He might be involved in the story in a way that either helps or hinders the situation where Naruto hurts Kakashi. For example, if Sasuke tries to stop Naruto from doing something stupid that would hurt Kakashi.
Another reason could be related to Kakashi's past actions. If there are some secrets or things Kakashi did in the past that come to light and Naruto doesn't like them, he could snap at Kakashi. Naruto has a strong sense of justice and loyalty, so if he feels betrayed or misled by Kakashi, his reaction could be to snap.
It can create more excitement. Fans get to see a beloved character engage with fan - made content, which makes them feel more involved. It might also inspire new fanfiction to be written, as people see the potential for their work to be 'highlighted' in this way.
One possible reason could be that Kakashi is preoccupied with other important matters in the village. Maybe he has some top - secret missions or tasks assigned by the Hokage that demand his full attention, leaving him with little time to focus on Naruto.
One reason could be that he sees Naruto as a younger version of himself in some ways. Both have had their share of loneliness and challenges growing up. Kakashi may want to prevent Naruto from going through the same pain he did. Another reason is Naruto's importance to the village. As the jinchuriki of the Nine - Tails, Naruto is a key asset. Kakashi, being a loyal ninja, wants to protect the village's future which is tied to Naruto.
Sakura could also be present. She might be observing the training and learning some things herself, or she could be helping Kakashi in some way, like providing medical support during intense training exercises.