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Throughout the entire series, Victor is depicted showing his affection for Yuuri very publicly.
A few examples of this being...
When Victor hugged Yuuri from behind in front of the entire audience at the CSK championship.
When Victor got completely drunk and started hanging off of Yuuri in front of everyone before the Cup of China, and had absolutely no complaints about the picture Phichit posted of them.
When Victor wrapped his arms around Yuuri's waist while resting his head on his shoulder in front of the other skaters at the Cup of China
When Victor kissed him in front of thousands of people on live television at the Cup of China.
When Victor publicly announced their engagement the rest of the gpf skaters + the others around them who were also there to witness it in Barcelona.
Must I go on?
Moral of the story is, he's so proud of being able to love Yuuri that he doesn't even care about his title as the 'living legend of figure skating' who went off to coach some 'dime a dozen' Japanese figure skater. He doesn't let the possibility of backlash affect the way he shows his partner his love, and he doesn't care about what anyone else has to say about their relationship but the two of them, and I think that's such an amazingly healthy aspect about their relationship that just isn't talked about enough.
It says so much about Victor as a character. It shows that Yuuri is so much more than just his student, fellow competitor, or fan. He sees Yuuri as his equal, and as someone worthy of showing off and taking care of. He loves Yuuri enough to put him above his title, and to sacrifice his competitive skating career for him. He refers to Yuuri as his life and his love, and was absolutely devastated at the idea of Yuuri not needing him as his coach anymore.
He loves Yuuri more than words, and everyday I grow even more upset that we never got to see more of them.