Izuku Yagi is a kind boy, he loves to hang around his mum, who he loves with most (not izuku X inko) and helps her and others whenever he can.
That all changed when he was diagnosed quirkless, he wanted to be with his mum a lot, but she started to grow distant from him over time, eventually neglecting him like his step-father did pretty much straight away.
But in reality izuku is just a late bloomer, izuku has a very powerful fire quirk which he got from his father who passed away before he was born.
He pretty much has Chong Jim ins powers from solo leveling, he will train in martial arts and use his fire with it as well.
Will have the same personality as him as well, and will be a great leader of his guild which he started when he is 16, which he did instead of going to UA.
He will be stronger and can do s rank dungeons alone, he will be doing dungeons from a young age, and will have collected a lot of money because of it, which is how he can afford a guild at just 16.
Dungeons are a thing, obviously, so the hunter profession is also a thing, izuku will be one of the strongest hunters, and will work with the other S-class hunters.
Hunter and heros are pretty much the same thing, and are seen as such, but they fight different enemies. People look up to both professions, but the hunter profession is more accepting for the quirk someone has, and doesn't really care what it is, unlike heros who are more biased for "heroic" quirks.
Izuku doesn't like his past, but will put up with them if needed.
He will solo jeju island, after he gets permission from the Korean association, and well as the Japanese. He will fight beru, but he uses his fire in a way to enhance his abilities and he can go toe to toe with beru, but over powers him because of his fire.
Will look like image above