Harry's sense of justice would still be there but might be expressed differently. As Jon Snow, he'd be more likely to take matters into his own hands in a more direct way. For instance, instead of relying on the Ministry of Magic's laws like he did in the wizarding world, he'd be more of a lone wolf in Westeros, making his own decisions on what's right and wrong. He'd also probably be more reserved in showing his emotions, as Jon Snow was not one to wear his heart on his sleeve.
Well, Harry was used to having friends and a support system at Hogwarts. As Jon Snow, he'd be more solitary. This could make his personality more self - reliant. He might not trust as easily as he did in his previous life. Also, his loyalty would be more complex. In the wizarding world, he was loyal to his friends and Dumbledore's cause. As Jon Snow, his loyalty would be divided between the Night's Watch, his family (both the Starks and his new - found relations), and the greater good of Westeros. His experiences as Harry Potter would still influence him, but the harsh environment of Westeros would shape him into a more hardened and perhaps more calculating individual.
He'd probably be more timid at first. Tsuna is known for being a bit of a wimp in the beginning, and even though Harry has his own insecurities, being in a new world and a new body might make him even more hesitant. But as he gets used to things, his brave side from his past life would start to show more. He'd still have that sense of justice and a willingness to protect his friends, just like he did in the Harry Potter world.
His confidence would still be there but might be a bit more subdued as he adjusts to the new world. He's used to being a big shot among demigods, but in Hogwarts, he has to start from scratch.
He might become more outgoing. In the Naruto world, the culture is different from the rather formal and often secluded wizarding world. Harry would be exposed to a more boisterous and open - minded society, so he could become more extroverted and less reserved.
One possibility could be that he would bring his magical skills into the world of Westeros. He might use magic secretly to gain an edge in battles, like creating distractions or protecting himself and his allies. For example, he could use a shielding charm during a skirmish with the wildlings.
His personality might become more reserved. Growing up with a Shinigami would expose him to different values than the normal wizarding ones. He might not be as quick to trust or be as carefree as in the original story. Instead, he'd be more cautious and perhaps a bit more brooding, always thinking about the deeper meaning of things and the consequences of actions.
His personality would be greatly influenced by the forest environment. He would be more wild and free - spirited. Growing up away from the constraints of wizarding society, he would have a different set of values. He would be more empathetic towards animals and nature, seeing them as equals rather than something to be used. He would also be more solitary at first, but once he meets others, he would form very deep and loyal relationships as he values connection on a deeper level.
His personality would likely become more conflicted. He has a good heart, but being around Bellatrix would make him question a lot of things. He might become more withdrawn as he tries to hide his true self from Bellatrix.
Harry Potter's personality would be drastically altered. Growing up with Azazel, he would lack the warm and nurturing environment that shaped the original Harry's kind - hearted nature. He might be more aloof and detached from others. Instead of being the hero who stands up for the weak, he could be more interested in using his power for his own gain. His sense of right and wrong would be blurred, influenced by Azazel's amoral ways, and he may not hesitate to use dark means to achieve his ends.
He might become more authoritative. Since he has the power of an emperor, he would need to make tough decisions and be more assertive in his commands.
His personality would likely become more self - assured. Growing up with the founders, he'd be constantly exposed to great magic and wisdom, which would boost his confidence. He might also be more independent as he'd be trained in a more solitary way by the founders.