Thats correct, the compressor is a single component who has no idea about the left and right side of the car. If you are blowing cold air in any of the vents, the compressor is working fine, unless ofcourse there is a leak somewhere and eventually the compressor will shut down, but that would not be because the compressor is bad.

Is the amount of air coming out from the warm vent the same as what is coming out from the cold vent ?

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Saumil S. Patel