C stands for “Common Mistake”
It stands to reason that they would be related.
They are not, at least not in the way that you think.
In both cases the C denotes a “common point” in the dual capacitor is is the common point between the fan capacitor (fan) and the compressor capacitor (herm). In the compressor it is the common point between the run and start windings (this is why R+C + S+C = R+S if you ohm a compressor)
The C terminal of a dual capacitor is actually fed from the OPPOSITE leg of power as the C terminal on the compressor. This is because you must power the start and run windings with the same leg and common with the other leg.
The way I always said it was “The same leg that feeds start feeds run” and the C terminal on a capacitor is actually the common feeds for the start winding of the compressor and fan (OPPOSITE side from the fan and herm plates on the capacitor)
So comp terminals
C goes to one leg
R goes to the other
S goes to the the terminal on a capacitor with the other side of that capacitor going to the same leg that feeds R.
C what I’m saying?