The reciprocity agreement does not apply to Ohio local city taxes. You are not required to file a local city return, but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount.
Michigan will give you credit for that on the MI state return, but you have to attach a copy of the city return. So, even though the Ohio city does not require you to file a return, you have to file one to get the MI credit, even though there will be no tax due or refunded from the Ohio city.
