Batman's identity would transform significantly. As a lantern, he'd have to balance his Gotham - based values with the values of the Lantern Corps. His identity as a detective might take a backseat sometimes as he's now dealing with intergalactic threats. His costume could also change to reflect his new status. He might add elements of the lantern uniform to his traditional Batman suit. And his relationships with his traditional enemies like the Joker might be put on hold as he has to deal with new threats from the Lantern universe.
Batman's character development in such a fanfiction is multi - faceted. Firstly, the power of the Lantern ring challenges his self - reliance. He has to rely on the power of the ring as well as his own skills. Secondly, his relationships with other characters change. He may have to work with other Lanterns, and his interactions with his traditional allies in Gotham like Robin or Commissioner Gordon may be affected. His moral code may also be tested as he is exposed to new cultures and situations in the Lantern Corps. Overall, it's a transformation that forces him to reevaluate who he is and what he stands for.
His character might become more isolated. With the power of the Juubi, Naruto may feel that he's on a different level from his friends. He could also become more cautious as he has to control such a huge power.
His personality might become more aggressive. The demon nature could bring out his primal instincts, making him less patient and more prone to anger.
Naruto's personality would become more brooding. He would be less impulsive and more calculating, like Batman. He might not be as openly friendly as he was before, keeping his thoughts and plans more to himself.
His story changes a great deal. He can't rely on his powers to solve problems. For instance, in battles, he'd have to use normal weapons or his intellect. His relationship with Annabeth would also be different. They might not have the same kind of adventures together. He'd be more in danger from the mythical world since he's no longer a powerful demigod.
His relationships would surely be strained. His friends might be afraid of him or try to reason with him, but Naruto's violent nature could drive a wedge between them. For instance, Sakura might be horrified by his new - found aggression and try to stay away, which is a big change from their usual dynamic.
Naruto's character would gain a new sense of responsibility. As a Green Lantern, he'd have to abide by the Corps' rules and uphold justice on a much larger scale. This might make him more serious at times, but his core of being a fun - loving and determined person would still remain. He'd also learn to work in a team with other Green Lanterns, which could be different from his experience with his ninja teams as they come from diverse backgrounds.
His already positive personality would become even more so. He'd be like a beacon of hope. He might be more focused on bringing out the best in others, since that's what Blue Lanterns do.
He becomes more authoritative. Harry, who was once just a student and a hero, now has to make decisions for an entire kingdom. So he has to be more assertive and confident in his actions.
He might become more cynical. Instead of seeing the good in people like he used to, he starts to suspect everyone's motives. For example, he may no longer trust his friends easily.