Kosuke might be a character created specifically for this fanfic. He could be someone with a unique background, like perhaps he was once a disciple of a great ninja but then went on his own path. And now he sees potential in Naruto and decides to train him. He could have abilities that are different from the main characters in the original Naruto series, which is why he has something new to offer to Naruto in terms of training.
In this fanfic, Lilith could be an original character created by the author. She might be a powerful and mysterious figure with special abilities that make her suitable to train Naruto.
She could be someone with a special bloodline or ability. Maybe she has a unique form of chakra that gives her certain powers. In the context of training Naruto, her skills and knowledge are what make her important. She could be a mysterious figure who has been observing Naruto for a while and decides to take him under her wing to help him reach his full potential. For instance, she might have the ability to see the latent potential in others and wants to shape Naruto into a great ninja not just for his own sake, but also for the greater good of the ninja world.
Maybe Muu is a long - lost member of a powerful clan that has some special connection to Naruto's Uzumaki heritage. So he takes it upon himself to train Naruto to help him reach his full potential and also to preserve some ancient techniques or secrets related to their clans.
I'm not sure who the specific author of that fanfiction is. There are so many fanfiction writers out there that it's difficult to determine without more research.
Maybe Kakashi could be the one. He's always been Naruto's mentor in many ways. He has the knowledge and skills to train Naruto for such an unusual task. He could use his Sharingan to analyze the fanfiction and teach Naruto how to spot the dangerous elements within it.
Another option could be Itachi. Although Itachi is from a different village and has his own complex situation, in a fanfiction scenario, he could secretly train Naruto. Itachi has extremely powerful jutsus and a deep understanding of the Sharingan. Training under him would give Naruto a very different perspective on power and combat, not to mention some insights into the Uchiha clan.
It could be a rogue ninja from a forgotten clan. This ninja might have a vendetta against Naruto's village and has powers that are difficult to counter. So, Naruto has to train to kill him to protect everyone.
There could be new antagonists created specifically for this fanfic. Maybe some dark forces that were sealed away by Asura long ago are awakened. They would oppose Naruto because they see him as a threat to their resurrection or their evil plans.