If Naruto was raised by Kakashi in fanfiction, their relationship would be very different from the original. Kakashi could teach Naruto more about the Sharingan and its secrets since he has one. Naruto might also inherit some of Kakashi's reading habits and love for Icha - Icha Paradise, which could lead to some really funny moments.
He'd probably be more reserved. Kakashi is not as boisterous as some others, so Naruto might pick up on that. For instance, he might not shout as much about becoming Hokage right away.
Kakashi would probably focus more on stealth training. In the original, Naruto was more about big, flashy attacks. But Kakashi would teach Naruto how to move quietly and strike unexpectedly. He might also teach Naruto more about chakra control from the very beginning. In the original, Naruto struggled with this for a while. And Kakashi would surely introduce Naruto to some of his own unique jutsu earlier, like the Chidori, but with Naruto's own spin on it.
One possible plot could be that Naruto would grow up with more refined ninja skills. Kakashi could teach him his signature jutsus like the Chidori in a more in - depth way. Kurenai might focus on Naruto's genjutsu skills, which he lacks in the original. With their combined guidance, Naruto could become a more well - rounded ninja at an earlier age.
In such a fanfiction, Naruto might become more strategic from an early age. Kakashi's intelligent and cautious fighting style could influence Naruto. He might teach Naruto more about the Sharingan he got from Obito earlier, which could make Naruto's combat skills more diverse.
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.
One possible main plotline could be Naruto growing up with Kakashi's unique training methods. Since Kakashi is known for his strict and strategic training, Naruto might develop different skills compared to the original series. Another plotline could be how Naruto deals with the abandonment issues from his past while having Kakashi as a father figure. He could struggle with trust issues at first but gradually open up as Kakashi shows him unconditional love and support.
In such a fanfiction, the female Naruto's growth as a ninja under Kakashi's tutelage is a key element. Kakashi, with his vast knowledge and skills, would train her in various jutsus, but also in strategy and survival. The fanfic may also focus on how she uses her femininity as an advantage in the ninja world. For example, she might be more perceptive in some situations or better at undercover work. Her relationships with other characters in the village, such as her peers and the elders, would also be an important aspect as she tries to make her mark as a female ninja raised by Kakashi.
One interesting fanfic could be 'The Female Naruto's Journey with Kakashi'. In this, the female Naruto might face unique challenges different from the male version in the original series. For example, she may have to deal with different expectations from the villagers. Kakashi, being her mentor, would train her in ways that suit her strengths and weaknesses as a female ninja.
I'm not aware of any such fanfic. Naruto is a male character in the series, and such a concept goes against the natural order of the Naruto universe. Fanfics usually follow the general theme and characteristics of the original work, and this seems quite out of place.