The short answer is absolutely!
In Sun Tzu's conceptual terms, the two idea are not opposites, but complementary opposites. Things that seem like they are opposites, but which really are two sides of the same thing, like light and dark. One cannot exist without the other.
In other articles such as this one, I have described how cooperation requires competition before it can take place. However, in a similar way, competition or at least competitive battles, often require cooperation before they can take place.
The most basic illustration is a military battle. A battle does not take place if one army avoids the other. This is the basis for guerilla warfare. Major military battles occur because both armies choose to meet at a specific place at a specific time. In other words, they cooperate.
Both armies choose to meet at a given time and place because they both think that it is in their advantage to do so. Battle time and location are never chosen by one side. Even in situation like D-day. The Allies choose to invade at a certain time and place, but Germany also choose to defend at the time and place. Germany could have avoided a shore battle, where the Allies had the support ot their superior navy. They could have let the Allies land and met them further in land. They chose not to because they felt that defending the shore, before the Allies established a beachhead, would be easier.
They were wrong.
In a sense, the Germans made the same mistake the French did by relying on the Marginot Line. They tried to defend the entire coast with fixed placements, spreading their forces, rather than using their mobility to concentrate their forces at a time and place of their choosing.
However, the clearest form of cooperation to create competition is in the sports world. Before their can be an organized sport, they is the league that organizes the contests. Many forms of competition, even economic competition, require working out the rules that people will play by before hand.