The manager and captain of the newly-crowned Premier League champions have both turned their attentions to next season as they focus on "building a big club"
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The club finished 11 points behind rivals Manchester United last season as they failed to successfully defend their 2011-12 title under Roberto Mancini, and both men are determined not to repeat past mistakes.
"Well of course I think that big teams cannot be satisfied just with one title," Pellegrini told Sky Sports. "It’s very important to win one or two titles in a year, we must celebrate today and tomorrow and on Tuesday start to work for next season because this club and these players deserve more titles.
"It’s special, because my first year here in England is not easy to be accustomed to English football but I think I manage a great group of players, a great institution, and our great fans."
Kompany, who settled any lingering nerves by scoring his side's second goal shortly after half-time, backed up his manager's message and vowed to turn City into a "big" club.
"We're building a club, not just a team that wins trophies every now and again, we're building a club," he said.
"The past is the past but surely we'll have our place in the future. That's why I'm excited to be here, that's why every single year I have a feeling that we need to be better, and next year we have to be even better.
"This has to transpire through to the whole club. If we want to be a big club then this should be one of so many. It will take a lot of work but people are ready for it."
Kompany also insisted that he is realising his chilhood dreams as City captain: "The way we finished this season is a credit to how we've always prepared, every single game and how we've always believed in ourselves. [I am] extremely happy and this is for the fans.
"Forget about the money, forget about everything. As a kid you grow up and all your dream of is playing games, winning games and lifting trophies. That's what I've dreamed of. Not the rest, I've dreamed of lifting trophies. I feel like I'm achieving that dream by doing this."