What is a Campbell Hausfeld® Switch?

It's a replacement pressure switch for starting and stopping the motor. I searched for the part # and found a bunch of them on Ebay.

It's a pressure switch (kit).

It's probably a combination of pressure switch (turns power to the compressor on and off) and also unloader (bleeds off the high pressure air between the compressor and the input check valve).

The unloader is important. A compressor does not like to start, when there is still pressure in the discharge line. The motor takes too much amps, and really is tough on the motor.

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Agreed. If you pay close attention to a compressor operating you can hear the unloaded release the pressure from the line when the motor stops. You hear the short "Pssst". If the unloaded doesn't work the motor won't start .

You are correct. However, since some people here like things to be excessively precise, the compressor does not like to start with a sealed compressor stage. Piston, sliding vane, or screw. :-)

Centrifugal types are a bit more forgiving if they have metal housings. Technically these are more of a blower, but I digress.