A vehicle air-conditioning system and a method for operating that vehicle air-conditioning system in dependence on the difference between the temperature of the engine coolant at the inlet into the heating heat exchanger and the temperature of the air at the outlet out of the heating heat exchanger are provided. The measured or estimated temperatures of the coolant at the inlet into the heating heat exchanger and of the air at the outlet out of the heating heat exchanger are checked to determine whether their values indicate fault states of components of the air-conditioning system.