Because they are from different games, they never met in canon. Despite this, they share an interaction on the Ultimate Talent Development Plan, in which they both seem to get along very well and have great respect for each other.
Korekiyo raises the issue with Nagito asking him if he is going to support the athletes, Nagito says yes and that he is happy to be on time, Korekiyo comments on him enjoying watching the talents of others and that he was surprised that there was no football player there, and Nagito says that he just wanted to see the moment when everyone shines as symbols of hope. Later, Nagito apologizes for someone like him distracting Korekiyo from the game, where Korekiyo says he doesn't care, saying that all aspects of humanity are beautiful and that he is also watching him with great interest, and Nagito says he is happy for someone like it can be useful for someone like Korekiyo. Then Nagito spends time watching Korekiyo and the football game.
Both are very similar, they could be considered excluded people in their respective games, Korekiyo because they find him weird and scary and Nagito because of his problematic personality revealed after the first Class Trial, but later Korekiyo is also revealed to be a troublesome person for having killed too many women to be friends with her deceased sister. Both have criminal fixes, moreover, they showed great admiration for the ultimate high schoolers, but in Korekiyo's case he feels admiration for all of humanity.
The ship left due to the similarity of their personalities and their interactions in the Ultimate Talent Development Plan. But because they are from different games, the ship is a rarepair, being quite overshadowed by Komahina for Nagito and Amaguji for Korekiyo.
- Both were caught while planning an assassination, Nagito by Teruteru and Korekiyo by Angie.