Ohio Rev. Code § 307.153
A board of health of a city or general health district may enter into an agreement with the board of county commissioners of the county in which the health district is totally or partially located, and the board of county commissioners may enter into an agreement with the board of health, whereby the board of health undertakes, and is authorized by the board of county commissioners, to exercise any power, perform any function, or render any service, in behalf of the county commissioners, which the board of county commissioners may exercise, perform, or render.
Upon the execution of the agreement and within the limitations prescribed by it, the board of health may exercise the same powers as the board of county commissioners possesses with respect to the performance of any function or the rendering of any service, which, by such agreement, it undertakes to perform or render, and all powers necessary or incidental thereto, as amply as the powers are possessed and exercised by the board of county commissioners directly. Any agreement authorized by this section does not suspend the possession by the board of county commissioners of any power or function exercised or performed by the board of health in pursuance of the agreement. Nor shall the board of health, by virtue of any agreement entered into under this section, acquire any power to levy taxes in behalf of the board of county commissioners unless approved by a majority of the electors of the county.
Every agreement entered into under this section shall provide, either in specific terms or by prescribing a method for determining the amounts, for any payments to be made by the county commissioners into the health fund of the health district, in consideration of the performance of the agreement.
R.C. §307.153