"with cooperation" is an acceptable phrase and can be used in written English.
For example, "With cooperation, the task was completed easily and quickly."
I shall deal first with cooperation on process.
But with cooperation, those problems, too, can be solved.
Even with cooperation, Mr. Cox predicted that Mr. Gordon will spend many years in prison.
But even with cooperation from its subjects, Winthrop can face hurdles.
The answer was a Europe that replaced old enmities with cooperation, he decided.
— Rita Kramer, Midtown Dr. Falcocchio: Delivery times could be restricted only with cooperation from businesses receiving deliveries.
Microsoft, the government charged, denied rivals early access to its technology, while rewarding its industry friends with cooperation.
And the Kremlin indicated that it would sign three other agreements dealing with cooperation against terrorism, stabilizing world energy supplies and expanding cultural and governmental contacts.
With cooperation from his counterparts, the chairman of the South Shore listing service, Gerard Longo, gathered statistics from all three services.
Appealing to them with cooperation and reasoning, rather than 'axis of evil' talk, is a virtually risk-free proposition for the U.S".
Ecological symbioses between fungi and plants, or between bacteria and fish, pale into insignificance when compared with cooperation within a body.
