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They canonically have never met before.
Both Byakuya Togami and Nagito Komaeda are participants of the Killing School games and the rival character to the protagonists in Danganronpa. Incase of Byakuya being Makoto Naegi and for Nagito Hajime Hinata.
Byakuya is a cold and self-centered person who is very reserved. He is not interested in cooperating with others due to his own sense of self-superiority, and his opinion is that others are, on the whole, lesser than him. This sense of superiority and black-and-white worldview comes as a result of his upbringing; Byakuya had to prove his worth in battle against his siblings in order to earn his role in the family as heir-apparent, whereas all his other siblings who lost the battle were effectively ex-communicated from the family and cut off financially, deemed "unfit for the name of Togami". Because Byakuya was raised in such a highly competitive environment of "haves" and "have-nots", he thinks all people only think about themselves and he dismisses emotional ties as petty and unimportant, in his own words.
Nagito on the other hand appears to be somewhat out of touch with his own feelings. He tries to be polite, but he can be overly harsh and insensitive, often without realizing it. He has a lack of social awareness and insight into his own behavior. Because of this, he has problems understanding how his own behavior can affect other people and he may not understand or pick up on social hints. He struggles with expressing himself in a socially appropriate manner and this often leads to people misunderstanding what he's trying to say, and vice versa. During Chapter 1's additional dialogue, Nagito explains that he says dubious things without meaning to and that he realizes this, but can't stop it from happening. However, he also adds that he always sincerely means what he says.
However After Nagito discovered that he himself and his classmates were the Remnants of Despair, Nagito began to act arrogant, impatient, and disrespectful towards his former idols.
Komagami resulted as being a rare pair between Nagito Komaeda and Byakuya Togami. Komagami is easily being overshadowed by their more popular ships with the protagonists of their respective games, Komahina and Naegami. With that being said, there is also a polyship between the two and Kokichi Oma. Even though this ship is being overshadowed and considered a rare pair it is still being shipped by quite some people.
- Both are rivals of the protagonists.