Aiding, assisting, conspiring, helping, joining forces, uniting, collaborating, working together, working side by side...
have been cooperating with both.
"we've been cooperating with" is the correct one.
antonyms of cooperating are: blocking, delaying, stopping, handicapping, hindering, obstructing, impeding, and preventing.
Organization of Cooperating Autonomous Trade Unions was created in 1985.
by cooperating
Working together
It is usually cooperation or co-operation (working together).*Note that co-operation can mean simultaneous operationof two separate processes, such as administration or machinery.
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collaboration
This is an oxy-moron you cannot be cooperating involuntarily.
by cooperating to your teammates
The verb is cooperate (cooperates, cooperating, cooperated).