Proving Carnot efficiency is maximum and conditions

[C_Pu]

The standard proof to show carnot efficiency cannot be exceeded is to couple a supposedly more efficient engine to a carnot refrigerator, and show that it would violate second law. However, isn't it true that we can make the same argument with any arbitrary efficiency?

Some discussions on stackexchange regarding this topic say the key point is only carnot engine is reversible which makes this line of argument specific to carnot efficiency. They also suggested the two-isotherm two-adiabatic carnot cycle is only for easier conceptualization of a reversible cycle.

Can someone clarify all this confusion please?

 

[DrClaude]

However, isn't it true that we can make the same argument with any arbitrary efficiency?

Could you clarify what you mean by that? (And also what you mean by "Carnot efficiency.")