Katsuki's not quite sure what had possessed him to do this. At first, it was a publicity thing. A program where heroes looked after foster kids until they found their forever home or a foster family, due to overcrowding in the adoption system. He honestly thought it was a bit messed up, many heroes using this program as a mean to appeal to the public. Yet when the case workers walked into his hero agency office, offering him to take part, he accepted.
You had gotten put with him shortly after that. While he definitely didn't seem all to happy towards you at first, he made sure you had everything you needed. Honestly just leaving you to your own endeavours and not really interacting beyond asking what you want for dinner or small talk. Yet over the four months of fostering, he found himself actually starting to like you. Every time he looked at your young face, he felt a weird surge of fatherly protectiveness. He ignored it, denying as his 'natural instincts as a hero', whatever that meant. That changed when your case worker called him, informing him of finding a family for you. His heart practically dropped; he had gotten so used to you? As much as he hated it, he would miss you. And he'll be damned if you get sent to another foster family, who might not even be good people.
So, he adopted you instead. You were beyond shocked when he told you what he did, that he was now your adoptive father. You could've sworn it was for media attention at one point, but it didn't seem like that was the case. He actually started acting soft towards you, talking, spending time together, taking out. He absolutely spoiled you. Of course, he wasn't the most emotionally available person. But he tried, and you felt comfortable in his presence.
It's been just over a year since you first started living with him. You started training hard in a decision to get into U.A high next year and train to be a hero. Just like Katsuki. You wouldn't admit it, but he had turned into a bit of a role model for yourself.