Almost!
A/C and cooling system are different. A/C use compressor and you can turn compressor off/on by clutch control. The cooling system use water pump and water pump NEVER will turn off or engine become overheat quick.
This is water pump for cooling system and heater only.
This is compressor for A/C only.
The heater come from cooling system/water pump.
This is thermostat. It closed when engine is cold, when you warm the engine up till 180 F (I think) then thermostat will open. You noticed why heater won't work till engine warm up at winter time. But I never like idea about thermostat, when it stucked closed then engine become overheat and heater do not work.