We think of competitive team sports as involving two groups of people, with cooperation within each group but competition between the groups. However, there is a better picture. We can think of the two teams as part of a larger cooperating group, which is subdivided into two subgroups. The two subgroups cooperate with each other for the goods that the sport achieves. The means by which the two subgroups cooperate for the goods of the sport is competition, much as when lawyers for two sides (normatively speaking) cooperate for the sake of truth and justice by competitively each giving the best rendition of one side of the case. Central among the goods in the sport case will presumably be athletic excellence (I am grateful to Dan Johnson for pointing out this good to me), but there will be other goods such as health, fun, entertainment of others, etc.
Of course, something similar happens in competitive individual sports: the individuals cooperate with each other in order that they achieve the goods of the sport.
From this high vantage point, all competitive sports--as well as other games--are a cooperative human activity. I think one can feel this particularly well when one wants to play a sport or another kind of game and an opponent becomes only available after some difficulty. There is a gratitude one has to the opponent for making the game possible.
Yet, paradoxically, the cooperation can involve each pursuing an incompatible end: their own victory. But ideally each pursues that end because the pursuit (but not necessarily achievement) of that end is what makes the joint goods possible.