system
Given the following post, respond as if you're commenting on Reddit:
human
(Batman) How can there be a Batlight on the top of the Gotham police department?

Why fund it? Who funded it?
How is the local government (let alone federal government) OK with there being a light to call a vigilante to do tax pay employees jobs?
gpt
Technically, Batman gave it to them in most continuities, and when he was considered an Urban Legend, it was just there to 'frighten' criminals.

That said, on New Earth, the government being involved with it was a large concern. So the Gotham City Police Department solved the problem by having a civilian hang out in Police HQ just to flick the switch. As long as an officer wasn't summoning Batman, there wasn't (as much) of a problem.

Also in New Earth, relying on a vigilante like Batman has given Commissioner Gordon a reputation as a complete joke in other cities. During the time Gotham was declared a No Man's Land, he was trying to get a job as a cop out of the city and cops basically started laughing at him as soon as he turned his back. Perhaps the only place that would accept him that wasn't Gotham would be Hub City. And you'd only go to Hub if you wanted to die.

And before you ask: Most other capes aren't vigilantes, but they're deputies. For example, Superman has special authority to arrest any criminal in Metropolis, provided he has evidence, and he can deputize other crime fighters to do the same. The Flash has a semi-official relationship with Keystone and Central City police departments, and he was paid to be an official superhero in California for a time. And the Justice Society technically still has the authority given to them by the President of the United States of America.