As an introductory physics student (independently studying before my return to official classes this summer), I am given a "proof" that the Carnot engine has the maximum efficiency possible by way of a diagram in which, assuming a more efficient engine whose work is fed into a reverse Carnot engine, we arrive at the contradiction of a net result of 100% efficient work because it is assumed the engine feeds, and the refrigerator (i.e. Carnot engine in reverse) takes, equal amounts into the cold reservoir. Having done research online, I have seen similar proofs in which the net effect is a heat transfer from a colder to a warmer body, but the main point is that, if there were a more efficient engine, we would violate the Second Law of Thermodynamics by feeding one into the other.
However, all of these "proofs" work equally well for ANY engine and refrigerator hooked up in tandem, one being more efficient than the other. I have looked everywhere for some sort of address on what seems to be an obvious oversight, yet cannot find anything except the occasional hand-waving.
I can "see" that a Carnot engine would be the most efficient based on complete reversibility, but I struggle with this manifestation of the proof (unfortunately, the only one I am advanced enough to understand, more than likely) because of what seems to be underlying assumptions/knowledge not addressed in any explicit statement I can find.
Can someone please clarify this proof for me, specifically addressing why it would not also apply to any engine/refrigerator in tandem where one is more efficient than the other? As I said, I'm sure there is a more advanced proof that improves the rigor, but I might not be ready for it, and I am hoping to get clarification on this specific proof.
Any help is MUCH appreciated. Thank you!