At some point Katsuki and Yukari went home together while having conversations about heroes, video games and fighting among others things.
While spending time during the Forest Training Camp Arc; for example when cooking together, they held various conversations and realized that they share the same opinion more often than expected. Both of them think that an hero should always give his all and never do half-hearted things. Nevertheless Yukari is not scared of disagreeing with Katsuki when they do think differently about something, always being confident enough to state her own opinion.

When Katsuki got abducted by the League of Villains during the Hideout Raid Arc Yukari immediately joined Eijirou, Izuku and the others to help them rescuing him. Her reason was that one can not always rely on the pros and that she is unable to just do nothing. After Yukari saw that Katsuki was safe, she felt beyond relieved, not even bothering hiding it. Because of this event the girl also realized how scared she was about losing him.

Katsuki accepted that he liked spending time with Yukari. Due to her feeling comfortable around Katsuki and also trusting him, she started to be more genuine when they were alone. Hiding less behind her jokester-attitude and being her true self instead, Yukari stopped acting as if she was carefree and overly bubbly. Katsuki would notice these things and appreciate it, obviously preferring this side of hers. Thus, he would get angry and frustrated whenever Yukari starts acting like a jokester again, causing him to start an argument.

Despite Katsukis aggressiveness and harsh words he was able to reach out to Yukari. Realizing that she cannot live her acts up, she decided to become more honest with herself instead of always bottling her emotions up. It can be said that Katsuki played a major role in helping Yukari with her inner conflicts, but it is still unclear if he did so due to care or annoyance. At this time Yukari already developed a deeper crush on Katsuki. On the other hand his feelings remain uncertain, but he slowly seems to care more about the pink-haired girl.

Will be added soon

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⇀ End of Season Four

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They only start dating later on, after Katsuki realized that he developed a crush on Yukari and tried to accept that he is attracted to her -- which only happens at the end of their second year at U.A. or even later. Their relationship is casual and not typically romantic. Katsuki appreciates that Yukari is easy-going and doesn't make things complicated for him. She just tells him whenever something is the matter without hesitating. Their 'dates' contain training- and video game sessions, going out to eat, making fun of basically anything possible or just lazing around and relaxing together.

Even after becoming a couple Yukaris jokes and teasing never stopped. Though with the time she became a bit more careful with that to say to Katsuki while in serious situations she avoids teasing him. Katsuki knows very well that Yukari has deep respect for him. After all she proved it many times and still does when it's necessary. This is also the reason why he becomes calmer around her and doesn't mind having her support. Nevertheless this doesn't mean that Katsuki will stop getting angry at her. They still bicker and argue, but it's never anything what would really hurt either of them.

» Yukari genuinely wants to make Katsuki smile or laugh and always tries her best to do so. Usually it's not working out but when it does work out Yukari is even happier about it, causing Katsuki to ask what's wrong with her and getting confused.

» Once Katsuki told Yukari to "go to hell". Yukari grinned in amusement afterwards and replied: "Nah, it's too early to go home yet."

» Yukari saved Katsuki as "Mr. BOMBastic" in her mobile phone's contact list and also used the song as his ringtone, because she thought it would be hilarious. Once Katsuki called her after Yukari lost her phone - and he was not amused at all.
Later on, Yukari put the song "Die Motherfucker Die" from the band Dope as his ringtone, because she thinks that song fits him perfectly. Katsuki seems to be fine with it.

» Whenever Katsuki is complaining about something Yukari will listen to him no matter how often he curses or insults others while doing so.

» Yukari likes to practice fighting with him, as she knows that he will always give his best and not go easy on her. She feels like both of them can benefit from it when training together.

» They like compete with each other; for example when playing video games, and try to beat each others highscores. One evening Yukari randomly sent a SMS to him where she wrote that she beat his highscore. Katsuki could absolutely not accept that defeat so he immediately got out of his bed and walked to her place. He bursted into Yukaris house, slammed the door to her room open, went straightly to the Playstation to play the game (while pouting very angrily) and spent the entire night there just to beat Yukari's highscore so he could be number one again. Yukari sat on her bed while watching him amused and eating popcorn, feeling super entertained.

» Yukari once sang out loud when she thought that she was alone in the dorms. However, when she strolled through the living-room while singing loudly she finally realized that she wasn't alone. Katsuki was leaning against the wall and looked up to see where the voice came from. Being embarrassed, Yukari immediately stopped singing and looked at the boy with the most irritated face ever. Katsuki couldn't help it and had to laugh because of her expression since he never saw her like this and it was truly satisfying to see Yukari being absolutely irritated for once. Nonetheless, Yukaris confusion didn't last for a long time since she quickly came up with a comeback.
Their conversation basically went like this:

Yukari: "Oh, guess I finally made you laugh~ See? I'm not a bad comedian at all" to which Katsuki replied with: "Shut your damn mouth. And so what if you made me laugh? Do you want a medal now?"
Yukaris answer was: "Nah, a trophy would be better." Thereafter Katsuki was the irritated one again.

» More to be added