A signal from the temperature sensor on the engine.
Depends on your definition of safe.
The engine's thermostat controls the engine coolant temperature. The temperature control inside for the heater controls the cabin heat.
As a "heat engine", a car engine must rid itself of heat to continue. Efficiency of the heat engine depends on the difference in temperature; therefore, as the temperature outside reaches the temperature inside, the engine stops.
A used car engine is safe if it was worked on by a certified professional. That way you know you will get quality added to the engine for optimal performance.
It is safe to drive you car for at least one mile after the check engine light comes on.
Mostly by engine temperature but also the sound of the engine.
If the engine temperature gets too hot and the engine fail - safe cooling mode is activated the engine will eventually shut off
If this is about a car: the heater in a car is basically another radiator, but this one sitting inside the car. When you turn the heater on you draw heat from the engine and into the car. If you aren't running the car hard enough the temperature in the engine will drop.
"Engine coolant temperature" sensor
We have no temperature gauge in our car. Didn't come equipped with one. It probably has a warning light that would illuminate if the engine was getting to hot.
It really depends on what the cause of the check engine light is.