The award of the contract to run that particular collection of routes is ostensibly free-market. Once it's awarded, they hold a monopoly for the duration of the contract though.

You asked how profits are extracted, and I answered your question. Now you're bringing up monopoly vs free market, which has nothing to do with the question.

Trains are a natural monopoly, that's why lassiez-faire policies don't work.

Nothing about a free market precludes a monopoly from existing. I have no idea why you would think that was the case.

> it's a monopoly, it's not a free market, by definition

It is a free market, you are free to open your own water company and build your own water pipes or deliver water in bottles. No one is stopping you

Actually, the very widely replicated experience of most challenger utilities is that the entrenched utility will stop you, sometimes in collusion with the government or municipalities. Try getting permits to dig up the road to put your competing pipes alongside the current monopolists.

In Britain, British Telecom spent around 20 years making it impossible for other players to supply an internet service to consumers. They used various anti-competitive practices and had very good capture of the largely toothless regulator OfCom. This is all documented.