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Climber's companion putting cold water on fire (3,3)

Ross

I believe the answer is:

ice axe

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'climber's companion' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.

'cold water on fire' is the wordplay.
'cold water' becomes 'ice'.
'on' is a charade indicator (letters next to each other) (in a down clue, letters appear on others).
'fire' becomes 'axe' (I've seen this in another clue).
'ice'+'axe'='ICE-AXE'

'putting' is the link.

Can you help me to learn more?

(Other definitions for ice axe that I've seen before include "Climber's tool" , "which a climber may use" , "Mountaineer's equipment" , "SW London area" , "Mountaineer's tool" .)

I've seen this clue in The Guardian.
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