The reason people ship this is because at first, Nagito worries about Mikan, as he scolds Hajime after thinking he bullied her and asks if she's okay after having one of her many exaggerated falls. He is also very humbled during the first trial when he mistakenly assumed that Mikan complimented his appearance when she pointed out that there wasn't any blood on him.
After Nagito caught the Despair Disease, Mikan took care of him along with the others. However, she also caught the disease, and despite his terrible condition, he noticed that she had fallen into despair. Because of this, and the murders she committed for the sake of despair, Nagito lost all respect for her and considered her an enemy, worthy of the name "Ultimate Despair". Unlike the others, he wasn't fooled by her actions during the trial and he bitterly refused to forgive her.
However, Mikan told Nagito that he's wrong and claimed that she had killed for "her beloved". Somewhat mockingly, she said that Nagito doesn't understand because he doesn't have anyone to love and isn't accepted by anyone, leaving him at a loss for words.
Later though, after Nagito and Mikan awaken from their comatose state, he doesn't seem to hold any more hatred towards her. This probably means that Nagito's original relationship with Mikan, which was mostly him not paying attention to her, although thinking her talent is amazing, has returned.
In Nagito's free time events, he used "Meeting a nice nurse after getting into a car crash" to explain his luck cycle, however, it's not confirmed that he was actually talking about Mikan.
In Danganronpa S: Ultimate Summer Camp, Nagito reassures Mikan when she worries that Monaca hates her, saying that it is likely because of her medical knowledge. He assures her that she hasn't done anything wrong and advises her not to push too hard.
- Mikan – “Well...Even if that's true, I feel Nagito looks...you know...clean...”
- Nagito – “Nobody's ever complimented me on my appearance before! Not even my own mother!”
- Mikan – “Nyehhh...I'm not talking about your looks!”
- — Mikan and Nagito while talking about Nagito probably not being the culprit, during the first Class Trial in Danganronpa 2.
- “If she ended up like that just from "remembering"... Then all along, Mikan was never worth being a Symbol of Hope. Even if her words were influenced by the Despair Disease, if the disease turned her into that... If she lost herself to a simple disease, she was definitely unworthy of being a Symbol of Hope. So...let's just move on. C'mon, let's take all this despair and change it into hope!”
- — Nagito losing his respect for Mikan after her Despair persona is revealed after the third Class Trial.
Fans ship Komamiki because how similar they are when expressing hate towards themselves (Nagito believing his talent is useless and not worthy of being around the ultimates and Mikan believing herself to be a mistake and constantly asking for forgiveness) and believe the two can console each other and even helping each other learn to love themselves. Their status as loners with no friends in the past and very few friends among their classmates also helped. Their dynamics of the "Medic Girl" and "Ill Boy" in Chapter 3 helped to fuel the ship further.
- Mikan's relationship with Nagito on Danganronpa Wiki
- Nagito's relationship with Mikan on Danganronpa Wiki
- They stand next-to-next during the Class Trials in Super Danganronpa 2: Goodbye Despair.
- Komahinamiki refers to the ship between the two and Hajime Hinata