Consider the two statements below:
(C1) It is impossible for a cyclic engine/heat-pump to transfer heat from a colder body to a hotter body without any third work/heat channel.
(C2) It is impossible for a cyclic engine/heat-pump to transfer heat from a colder heat reservoir to a hotter heat reservoir without any third work/heat channel.
Clearly (C1) implies (C2) almost automatically, because a heat reservoir is counted as a body. However, does (C2) imply (C1)? If so, what is the proof that (C2) implies (C1)? Is there some extra fact that is needed?
I find this an important question, because the Kelvin-Planck statement (as I know it) involves heat reservoirs. You can easily derive (C2) using the Kelvin-Planck statement, but the Clausius statement is usually stated as (C1).
I've seen only one link that acknowledges the difference between the two, and it is this question: Clausius statement of the 2nd Law. However, that question doesn't ask what I'm wondering about.
Edit: For clarification, let's say we negate (C1) (to get a negation of (C2)). Then there exists at least one situation where you have heat flowing from a cold body to a hot body.
My question is, how does the existence of this one situation imply that there exists a situation where heat flows from a cold reservoir to a hot reservoir?