Mich. Comp. Laws § 90.8

Sec. 8. Every city shall be allowed the use of the jail of the county in which it is located, for the confinement of all persons liable to imprisonment under the ordinances thereof, or under any of the provisions of this act; and any person so liable to imprisonment may be sentenced to, and committed to imprisonment, in such county jail or in the city prison, or other place of confinement provided by the city, or authorized by law, and the sheriff or other keeper of such jail, or other place of confinement or imprisonment, shall receive and safely keep any person committed thereto as aforesaid, until lawfully discharged.