Ohio Rev. Code Ann. § 5557.02

The board of county commissioners may construct a proposed road improvement into, within, or through a municipal corporation, when the consent of the legislative authority of such municipal corporation has been first obtained. Such consent shall be evidenced by the proper action of the legislative authority, entered upon its records, and the legislative authority may assume and pay such proportion of the cost of that part of the proposed improvement within the municipal corporation as agreed upon between the board and legislative authority. If no part of the cost of the proposed improvement is assumed by the municipal corporation, no action on its part, other than the giving of the consent above referred to, shall be necessary, and all other proceedings in connection with such improvement shall be conducted in the same manner as though the improvement were situated wholly outside a municipal corporation.