Coleman has revealed how, in the dressing room post-match, Koeman was unhappy that the Toffees, who were cruising 3-0 at half-time, let their opponents back in the game before running out convincing 6-3 winners with a late rally.
And the Republic of Ireland captain believes a measure of just how much progress Everton are making in such a short space of time is that they didn’t make the same mistake in the 3-0 victory over West Bromwich Albion.
Last Saturday, the Blues were commanding throughout and ensured there was no way back for the Baggies as soon as Kevin Mirallas got the 39th-minute breakthrough.
“We learned a lot from that Bournemouth game,” Coleman told evertontv. “The manager had a word with us afterwards. He was glad we got the three points that day but he wasn’t happy with how the game unfolded. Against West Brom, we made sure they didn’t get back in the game.
“It’s something he has worked on from the start but when you do get a run of good results, it comes naturally. Everyone is enjoying what they are doing at the moment, which makes it easier.
“The manager is never completely happy, which is a good thing. He always sees something that needs to improve.
“I’m enjoying playing for him. He demands a lot from us every day and that’s what you want as a footballer.”
Last weekend's FA Cup quarter-final results means that should Everton finish in the Premier League position they currently occupy - seventh - it will be enough to secure a place in the third qualifying round of next season’s Europa League.
With a seven-point gap to eighth-placed West Brom, the Blues are in a strong position - albeit Coleman is looking at the teams ahead, not below.
“There’s a lot to be done,” the defender added. “As Everton fans and Everton players, we’ve got win on Saturday against Hull and look up from there.
“We’ve got to keep looking up the table. We know there’s a bit of a gap but you never know what can happen. We’ve just got to keep winning games.”