Kumble stepped down a couple of days back due to differences with the India captain.
Kohli said "he respects" Kumble's decision to quit and that there is highest regard for him as a player.
"Obviously Anil bhai has expressed his views and he has taken a decision to step out and we will respect that decision. It's something that has happened right after the tournament," Kohli said on the eve of the first ODI against the West Indies.
"One thing for sure is that I have had 11 press conferences that have happened during the Champions Trophy. We have created a culture over the last 3-4 years that whatever happens in the change room, we have tried to maintain the sanctity of the change room throughout. That is what the whole team believes in. For us that is paramount," Kohli stated.
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The skipper said that he and his players respect Kumble's achievements as a player.
He was asked about Kumble's approach while coaching and he stoically maintained his stance.
"Like I said, for me what's most important is to maintain the sanctity of the dressing-room and what happens in the change-room is something that's very sacred and private to all of us.
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