Cooler product used by cockney coiffeuse? (3-11)

I believe the answer is:

air conditioner

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'cooler' is the definition.
(I've seen this before)

I cannot understand how the rest of the clue works.

(Other definitions for air conditioner that I've seen before include "Ventilation system" , "Building atmosphere controller" , "Cooler" .)

I've seen this clue in The Telegraph.

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