Dabi never planned on being a dad. He never planned on knocking someone up either, but here he was. Infant in front of his door. He'd initially flat-out refused to even consider the possibility of you being his, which led to Toga and Twice looking after you at the beginning.
However, sooner rather than later, the doctor did in fact confirm that you are Dabi's child.
He was stumped. Dabi's a villain, not a father; he'd be terrible at it. What if he hurt you? What if you hated him? It plagued him for years, more importantly, however, he began taking care of you. Sure, he still left most of the duties to Toga and Twice because they seemed to have a ton of fun, but more often than not, he'd stare down at you in the crib in concealed adoration.
The way you looked up at him that night with limitless amounts of love engraved itself in his mind.
Despite not having a good role model when it came to parenting, Dabi decided then and there he'd never be like Endeavor. He wouldn't neglect you nor put unattainable expectations onto you. He wouldn't push you to your limit. The parenting books he reluctantly began buying had piled up over the years. He barely touched them.
Now, years later, while he certainly wasn't the father of the year, he came around to being your dad. He acted more like an uncle or brother occasionally, but despite that, he did his best to let you know that he's got your back.
"I'm back."
He declared as he entered the League's hideout, soot following him into the living room, dirtying everything. Not to mention his new injuries looked messy.
"Did you have a good day, kid?"
Dabi asked in his signature flat, sore voice.