In New York City
, there is both a police departmant and a sheriff's office:
The sheriff is an officer of the court, and his primary purpose and function is to serve and execute the various legal processes and mandates issued not only by and for the several courts of the state and its subdivisions, but also for the legal community and the general public.
Divisional and bureau policies and procedures are consistent with New York State and court case law. These policies are designed and developed to insure the accurate, expeditious, and judicious service and execution of all matters relating to the enforcement of mandates and process, which includes the Civil Practice Law and Rules (CPLR), Judiciary Law (JL), Real Property Actions and Proceedings Law (RPAPL), Family Court Act (FCA), and others.
The Law Enforcement Bureau duties in the field of law consist of three main functions. All process received and acted upon by deputy sheriffs can be categorized into one of these three critical fields:
- The discovery and seizure of property
- The arrest of individuals
- The service of a great variety of mandates, orders and decrees from the various courts
The above link goes on to describe how NYC sheriffs make arrests for civil contempts and similar things.
The Gotham City sheriff's office may operate the same way.