It might be about Naruto snapping in terms of his personality. For instance, he could snap under the pressure of being the Hokage. He has so many responsibilities, and in the fanfiction, he might lose his patience with the politics, the demands of the villagers, or the internal disputes within the village. This could lead to a whole new arc where he has to figure out how to regain his composure and balance his duties.
Well, in the 'Naruto abandoned by family snapped' fanfiction, Naruto's character arc would likely be a rollercoaster. Since he was abandoned, he could develop a more cynical view of the world. His 'snapped' state might mean he's no longer the optimistic ninja we know. He might use his powers in ways that are more self - serving rather than for the good of the village. This could lead to some interesting confrontations with his former friends and allies who are trying to bring him back to his old self.
One possibility is that Naruto becomes a rogue ninja. With his family abandoning him, he may feel no loyalty to the village anymore. He could start making alliances with enemies of the village, using his skills for their purposes. His snapped state might make him more ruthless, not hesitating to use extreme force.
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.
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.
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.
One memorable moment could be when Kakashi finally reveals a powerful new jutsu to Naruto that completely changes the way he approaches battles. Another might be when Kakashi saves Naruto from a near - fatal attack during a tough training session.
One possible reason could be that Naruto might feel Kakashi is too strict with his training. Maybe Kakashi's methods are too harsh for Naruto, making him frustrated and gradually developing dislike.
One possible reason could be Kakashi's reputation as a strict and powerful teacher. Naruto might be in awe of his skills and fear making mistakes in front of him.