I am manager of team A. Before I became manager Steve (not his real name) belonged to team A and enjoyed the work.
Steve was moved to team B. He dislikes the work performed by team B.
Last week I discovered that, without discussing the matter with anyone, Steve was continuing to perform tasks for my team. He is friends with many people in the organisation, he did excellent work when on my team, and after moving he fostered connections rather than sever them.
I severed them.
But what I've not done is tell his manager that this occurred. It's not my job to point out to Steve's manager that they are failing to manage Steve effectively. To put it another way, if my team B colleague didn't work out over a period of weeks that Steve wasn't doing his job that's not my business.
Or is it.
Should I tell the other manager what happened?
Edit: I missed a crucial detail. His team - even with him added to it - remains severely under-resourced which is causing significant pain for stakeholders.