Ohio Rev. Code § 1905.35: Imprisonment.

Imprisonment under the ordinances of a municipal corporation shall be in the workhouse or other jail of the municipal corporation. Any municipal corporation not provided with a workhouse, or other jail, may, for the purpose of imprisonment, use the county jail, at the expense of the municipal corporation, until the municipal corporation is provided with a prison, house of correction, or workhouse. Persons so imprisoned in the county jail are under the charge of the sheriff. Such sheriff shall receive and hold such persons in the manner prescribed by the ordinances of the municipal corporation, until such persons are legally discharged.