In short, yes. A car's radiator fan should always start spinning once you've turned on the air conditioning unit. This is necessary, as your car's AC unit is reliant on the radiator fans working for its heat exchange process. Basically, it forces air to flow over the AC condenser, getting rid of excess heat away from the condenser. This allows your AC core to cool down, enabling you to enjoy cold air from the cabin. If the radiator fans didn't spin, the AC unit would remain warm, instead. In some cars, the AC unit's condenser unit might feature a dedicated fan, separate from the radiator fan. Oftentimes, the radiator fan turns on once the AC compressor is circulating refrigerant/freon throughout your car's A/C system.