The switch that turns your fan on if your coolant temperature is getting high is a coolant temperature switch. I don't have information on where that switch is on a Corolla 4A-F engine, but on a Celica 4A-FE engine, that switch is at the bottom of the radiator. If you disconnect the wiring harness from that coolant temperature switch, the fan should turn on at high speed. If your coolant hasn't reached that elevated temperature, the switch will not open, causing the fan to turn on. In normal driving, the fan will not turn on as the car is moving and has air going through the radiator cooling the coolant. If the car is parked with the engine running, there is no air going through the radiator to cool the coolant, so it is normal after idling while parked for 10-20 minutes for the fan to start cycling on & off. If yours does this, you don't have a problem to solve.