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Discover LudwigThe phrase "the cooperation" is correct and usable in written English.
It can be used when referring to a specific instance or type of collaboration between parties. Example: "The cooperation between the two companies has led to significant advancements in technology."
Dictionary
the cooperation
noun
The act of cooperating.
Exact(55)
The cooperation broke down.
The cooperation sometimes falters.
The cooperation comes from necessity.
(Greengrass secured the cooperation of them all).
But Agassi requires the cooperation of automakers.
We'd need the cooperation of the Chinese".
This task is impossible without the cooperation of Russia.
"The cooperation between our two countries is unprecedented," he said.
Foucault had the cooperation of this Age of the Engineer.
The cooperation vital to the whole programme would disappear.
"It requires the cooperation of all companies," Mr. Ronen said.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com