Nab criminal wearing soft fabric, one running the cooler (3,4)
I believe the answer is:
fan belt
'one running the cooler' is the definition.
(part of a cooling fan)
'nab criminal wearing soft fabric' is the wordplay.
'criminal' indicates anagramming the letters.
'wearing' means one lot of letters goes inside another.
'soft fabric' becomes 'felt' (I've seen this before).
'nab' is an anagram of 'anb'.
'anb' going inside 'felt' is 'FAN-BELT'.
(Other definitions for fan belt that I've seen before include "Drive to car radiator cooler and alternator" , "driver in engine" , "essential car component" , "Motor accessory" , "Band in car driving alternator" .)