This is an argument against specifically the notion of anarcho-capitalism, which is, in short, the idea that -all- government functions can and should be exported to private enterprise, including particularly that of police work.
Let's imagine an anarcho-capitalist society for a moment, shall we?
We have a nice pleasant neighborhood. Everybody except Sam subscribes to the local Hamsfield Policing Corporation; Sam, who lives in the corner lot, subscribes to the Unity Corporation, a massive policing corporation which services people all across the continent, and which is a popular choice for individuals living in areas with local policing corporations because it refuses to negotiate with them.
Tom comes to knock on Sam's door one day; they've had a longstanding argument about where exactly their property lines lay. Tom comes into the house, things get angry, and Sam shoots Tom.
Sam calls the Unity Corporation, who come out, investigate, and declare that it was self defense. Hamsfield Policing Corporation is called by Tom's wife, and Sam refuses to let them in; after talking to the neighbors, they conclude it was likely murder, and demand entry into Sam's home to investigate further.
Who has jurisdiction over Sam?
Either Unity hands Sam over to Hamsfield - which destroys its reputation and the very reason people subscribe to its services - or Hamsfield and Unity face off; most likely, Hamsfield, with more limited resources, will back down. But in either case, either Sam or Tom are not getting the service they paid for because there is no way to provide it for both.
There is no substantive way to resolve this issue, which is even assuming the police forces are acting in good faith and neither of them are favoring their subscribers over the neighbors. Anarcho-capitalism does not resolve the fundamental problems government exists to resolve; at best it creates a form of agency-facilitated vigilante justice.