Cooler that is cold inside (3)
I believe the answer is:
ice
'cooler' is the definition.
(something which cools)
'that is cold inside' is the wordplay.
'that is' becomes 'ie' (short for Latin 'id est', 'that is').
'cold' becomes 'C' (eg on taps).
'inside' means one lot of letters goes inside another.
'ie' enclosing 'c' is 'ICE'.
(Other definitions for ice that I've seen before include "Diamonds (informal)" , "Cold demeanour" , "Being feeble, can't cut any" , "Chill - harden" , "dessert" .)