Inko is the mother of Toshinori's protege and successor, Izuku. The two respect each other greatly, and Inko is well aware of how much her son has admired Toshinori since he was a young child. Although Inko knew that Toshinori was Izuku's mentor and teacher, they do not truly talk until the home visits after Toshinori fights All For One.
During the home visit, Inko initially refuses to let Izuku move into dorms at U.A., fearful of the danger her son is facing. Toshinori pleads with her, dropping into a bow on the ground and saying he promises to keep her son safe, even if it meant his own life being lost. Inko agrees, but only under the condition that Toshinori do his best to not only keep Izuku safe, but himself as well. Inko knows how important Toshinori has been to her son, and doesn't want her son to lose the hope and determination he has gained thanks to Toshinori. Toshinori agrees to this.
After Toshinori leaves, he talks to Izuku. He seems to be fond of Inko, and respects her greatly. He tells Izuku as much, stating she reminds him of his master, the previous holder of One for All, Nana Shimura.
Right after the 1-A parents' day in the School Briefs, which consisted on a masked villain that had captured the most beloved students' people and so the kids had to rescue them while they were on a tower with what they learned in the rescuing practice that Aizawa had teached them before, Inko apologizes to her son for scaring him, while she said that every parent was hired for that day, while Izuku reassured her that everything was fine. Before Inko left the place along with the parents, she approaches to [All Might], who was dressing as the masked villain, and she thanked him about back in the practice, while [All Might] said in response that he was glad that she got out of the place in only one piece. When Izuku and All Might where alone now, he thanks AM with a deep bow for saving his mother right during the rescue practice. All Might only responses back that it was only a bad habit coming from him, and that if he didn't save Inko Izuku would have gotten the whole work done himself. At the end of the story Izuku says for himself thath e would hold back his tears for only focusing on saving people.
This ship gained popularity after the home visit episode finally showing Toshinori and Inko truly interacting, and having a large mutual respect for each other. However, this ship has existed widely beyond that interaction, due to the absence of Izuku's biological father (and Inko's husband), Hisashi Midoriya, for the entire series, leaving Inko to face most of the troubles of Izuku's quirklessness and bullying on her own. Since Toshinori takes on a fatherly role towards Izuku after giving him his power, and has made vows to protect him, many fans like the idea of the three becoming a true family.
That being said, this ship is popular among those who enjoy the family ship ToshiDeku, as ToshInko tends to complete that family dynamic, with Toshinori truly becoming Izuku's father. Fan works tend to portray the two falling in love through their raising Izuku, finding solace in each other as Inko has long been away from her husband, and Toshinori is no longer the hero he once was. They often marry, making Toshinori Izuku's step-father. Other works for this ship include AUs where Toshinori and Inko have been married from the start, making Toshinori either Izuku's biological father instead of Hisashi, or his adoptive father, depending on when the two get together.
- Midoriya Family refers to the ship between Toshinori, Inko, and Izuku Midoriya