Ohio Rev. Code § 749.16
The legislative authority of a municipal corporation may, under agreement with a joint township hospital district board, or with the board of county commissioners, accept participation by such district or county in the erection or enlargement of a municipal hospital, or in the maintenance and operation of such a hospital, or both. Such agreement may provide for the amounts to be contributed by such district or county, as the case may be, for the construction and enlargement of such hospital, for its maintenance and operation, the rights and privileges to be enjoyed by the district and its inhabitants, or by the county and its inhabitants, by virtue of such contributions, and its rights of representation upon the municipal corporation's board of hospital commissioners or board of governors.
R.C. §749.16