Cheese is significantly more problematic than milk, because for cheese all Poskim agree that Chazal enacted a strict, absolute requirement for direct Jewish supervision since rennet from the stomach of a nevaila, an animal that was not slaughtered according to halacha, was typically used to congeal the milk into its solid state of cheese.
Even when it is known with certainty that no problematic ingredients are present, it is universally accepted that a Jew must be present. In addition to watching the process, the Shach and other Achronim require a Jew to actively and physically participate in adding the enzymes to the milk. Most mehadrin hechsherim follow this opinion. The halacha for the milk base is identical for cheese as it is for liquid milk, and one who relies on Reb Moshe's heter for Cholov Stam may use cheese derived from Cholov Stam, provided the cheese-making process was directly under Jewish inspection.