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Izuku has always admired All Might since he was a small child and wanted to be a hero just like him. At their first meeting, All Might thought that Izuku was just a fan, but he started to acknowledge him when he jumped to save Katsuki from the Slime Villain. From that moment, he was touched by Izuku's bravery, how he puts himself in danger for the sake of others despite not having a Quirk. He then decided he would pass his powers to Izuku, feeling that he is the only one worthy enough to inherit his strength, and they started to share a feeling of mutual respect for each other. He himself even mentioned that he has no regrets passing One For All to Izuku.
All Might is his confidante, mentor, friend in and out of school, and they share a mutual respect for each other's heroism ideals. Izuku's admiration and respect for All Might is so great that he had trouble trying to attack All Might during the End of Term Test, and when he had no choice but to attack All Might with Katsuki's Grenade Bracer, Izuku apologized.
Toshinori started to open up to Izuku as he sees his growth, going as far as to reveal personal details about his past such as how he was Quirkless, his fight with All For One and his scarring as a result of that battle.
He continues to express his pride in his growth as a hero.
In the first movie My Hero Academia: Two Heroes, Toshinori reives an invite to I-Island and with it being summer break he decides to bring Izuku along as his guest. Izuku is honoured that he gets visited to I-Island with his idol and mentor, and the way that All Might tells him that their bond with One For All is thicker than blood. Despite that touching fact, he asks Izuku to not share the fact that they came to the island together, due to the secrecy surrounding One For All. After All Might explains the island to Izuku, he takes him to see his friend, David Shield and his daughter, Melissa. Izuku fan boys out when he meets the very man who designed all of All Might's hero clothes and is glad to have gained a new friend in Melissa, who sees All Might has her uncle. When the island gets attacked by villains and All Might gets restrained by them, Izuku knew that the only way to help him is for him and his friends to regain control of the island's security system. All Might didn't want Izuku to do so, but he was also pleased by the fact that his student is acting like a hero as it reminds him why he chose Izuku. Once All Might was freed and the other pro heroes took care of the villains at the party, he goes to help Izuku who is trying to save David Shield from Wolfram.
After the fighting of the Paranormal Liberation War arc, Izuku is taken to the hospital and Toshinori pays him a visit. He is saddened that his student's all deal has heavily injured him and is currently in a coma.
It is related to their mentor/student relationship as well as a father/son relationship. It is very common to find stories where Yagi marries Midoriya's mother, Inko, as well as ones where Midoriya is All Might's biological son. Plots where Midoriya's mom dies and Yagi Toshinori ends up adopting him are also frequent.
The ship became extremely popular after Todoroki kept making comments how Midoriya was All Might's "secret-love child." He made the connections after noticing how similar Midoriya and All Might's quirk were, unbeknownst of him, that was because the quirks were one and the same.
On AO3, DadMight is the second most written ship for Yagi and the sixth most written for Midoriya. The most kudoed work in the My Hero Academia tag on AO3 is a DadMight piece.
- Midoriya Family refers to the ship between Toshinori, Izuku and Inko Midoriya