In some fanfictions, it could be due to a special jutsu gone wrong. Maybe a jutsu that Kakashi was using to transfer some of his chakra to Naruto for a mission accidentally transferred his Sharingan power as well.
The fanfiction might show Kakashi growing in terms of his understanding of the Sharingan's true nature. As he delves deeper into the abilities of the eternal Mangekyou Sharingan, he could gain a broader perspective on the Uchiha and their powers, which in turn enriches his own character development. He may start to see things from an Uchiha - like point of view, which was previously foreign to him. Also, the new power might attract new enemies or allies, and through these interactions, Kakashi will have to adapt and grow. For example, if new enemies target him because of his powerful Sharingan, he'll need to find ways to outwit them and protect those he cares about. This will surely contribute to his character growth.
Well, Naruto already has his own powerful abilities like the Nine - Tails chakra. But getting Kakashi's Sharingan could enhance his perception skills. He'd be able to see attacks better and react faster. Maybe he could even copy some jutsu more easily, just like Kakashi used the Sharingan to copy many jutsu in the original series.
One possible reason could be that Kakashi is in a life - threatening situation and realizes that Naruto is the best candidate to inherit his Sharingan. He knows Naruto's potential and his desire to protect the village. Another reason could be that Kakashi wants to see how Naruto would use the Sharingan in combination with his own unique abilities. He's curious about the new powers and techniques that Naruto could create with it. Also, it could be a way for Kakashi to show his complete trust in Naruto, as the Sharingan is a very precious and powerful gift.
Having Kakashi's Sharingan would force Naruto to mature faster in terms of his strategic thinking. He can't just rely on his Nine - Tails chakra and his normal combat style. The Sharingan would expose him to a whole new level of ninja tactics. This would also impact his relationships. His friends might look at him differently, and he might feel a bit isolated as he tries to master this new power. Over time, though, it could make him a more well - rounded and respected leader in the shinobi world as he combines the Sharingan's abilities with his own strong will and values.
In this fanfiction, Kakashi could help Naruto in his training through mental conditioning as well. Naruto is often impulsive, so Kakashi may work on his patience and decision - making skills. He could use his Sharingan (if applicable in the fanfiction) to analyze Naruto's movements during training and give him detailed feedback. Moreover, Kakashi could introduce Naruto to different types of enemies and battle scenarios in a controlled training environment, which would broaden Naruto's understanding of combat and make him a more versatile ninja.
His relationship with Sasuke would become more complex. Sasuke, as an Uchiha, might feel threatened or intrigued by Naruto having the Sharingan. They could have a new level of rivalry or a new form of cooperation.
He often keeps a close eye on Naruto. In many fanfictions, Kakashi will be secretly observing Naruto during his training or missions, just to make sure he doesn't get into too much trouble.
One great fanfiction could be about Kakashi using his Sharingan to train Naruto in a special jutsu. Maybe he shows Naruto how to better perceive an opponent's moves by using the Sharingan's ability to analyze. This could lead to Naruto developing a new and powerful technique that combines his own chakra control with the insights from the Sharingan.
Naruto could show his 'like Kakashi' traits in his mentoring style in the fanfic. Kakashi is a great mentor to his students, and if Naruto is like him, he might start to take on apprentices and teach them in a patient and knowledgeable way, passing on not just ninja skills but also important life lessons.