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Oki! Ummm-Kaz first 

Kaz was born into a low income family in the shittier parts of Hosu City. He was born with two quirks, his wings and his fire. His dad was a thug and just generally an absolutely awful person who beat his wife and son, and Kaz consistently tried to use his wings to escape and fly away, thinking that a pro-hero would rescue him and take him away from this awful place. One day after a particularly troublesome escape attempt, his dad got fed up. He restrained his then 11 year old son, and using a meat cleaver chopped his wings off, putting an end to Kaz's dreams. Kaz's dad was caught and arrested by the pros three years after this, and his mom died of fever shortly after, unable to get medical attention in their area and without the meager income his dad provided. Alone, Kaz wandered the streets homeless for a while, committing petty crimes and fighting to stay alive, that is, until he was taken in by Overhaul and the Shie Hassaikai. Kaz worked with them for about two years, during which time he learned about Eri and their plans. Kaz hated the abuse they were putting her through, and secretly planned to free her, but being a low-level grunt meant that he could never get close. After the group was taken down, Kaz managed to escape capture and went back to being practically homeless, living in abandoned places and committing crimes to live. Kaz has a general disregard for society and really doesn't care what happens to it, but he has no interest in joining the LOV despite being repeatedly scouted by them, Dabi in particular-he doesn't really believe in committing crimes for chaos's sake. However, the fairly large scale of his crimes and the similarity between his power and Dabi's have garnered him unwanted attention and lead many to think he's working with the LOV and the Meta Liberation Army. He really fucking hates Dabi.