In deriving thermodynamic temperature scale, it uses the concept that heat transfer between two reservoirs is the function of the reservoirs' temperatures. i.e. Q1/Q2=Φ(T1,T2). And then further express that Φ(T1,T2)=ψ(T1)/ψ(T2). I have two questions, 1. Is it a hidden assumption that the function Φ doesn't change its form for different temperatures? i.e. for different reservoirs we just plug in different temperatures Φ(T1,T2), Φ(T3,T4) but it can never be some other functions for different temperatures. Why? 2. Why Φ(T1,T2) can be expressed as ψ(T1)/ψ(T2)? Any underlying assumptions?