The contactor is a $10 part that resides in the outdoor unit, along w/
the capacitor. The capacitor operates on a magnetic switch, which pulls
in when there is a call from the thermostat for A/C. I replaced my
contactor after my Nest threw a E103 err...
I agree that the Nest E103 error usually means the $10 contactor and not
the $200+ Nest has failed. Also check the wires that go to the contactor
in the outdoor unit for corrosion and loose and broken connectors, all
of which can throw a E103 error.