cooperate with (someone or something)
To work together with someone or something, usually to complete a particular task or goal. Getting my toddler to cooperate with me is often an impossible task. We are in no way blaming you for what happened, but you still have to cooperate with the police investigation.
cooperate with someone (on something)
and cooperate (with someone) on somethingto work together in harmony with someone on something. Please cooperate with me on this project. Can you cooperate on this with me? I hope we can cooperate on this.