Most of the physics textbooks I’ve read try to prove that the Carnot engine is the most efficient engine possible by showing that if you coupled a superefficient engine to a Carnot refrigerator, you’d get a perfect refrigerator, which violates the second law of thermodynamics.
An example from Physics for Scientists and Engineers: A Strategic Approach is shown below.
Another example is this video by Khan Academy.
What I do not understand is what about this refrigerator identifies it as as Carnot refrigerator. Couldn’t you take any refrigerator, including one whose coefficient of performance is less than that of a Carnot refrigerator, couple it to a heat engine that is more efficient than the refrigerator driven in reverse (i.e. its engine “counterpart”), and get the same result that violates the second law of thermodynamics?