Path one straightens out for a carriage (7)

I believe the answer is:

phaeton

I'm a little stuck... Click here to teach me more about this clue!

'carriage' is the definition.
(type of horse-drawn carriage)

'path one straightens out' is the wordplay.
'path one' can be anagrammed to 'PHAETON'.
However, I'm not clear how the anagram is indicated.

'for a' acts as a link.

Can you help me to learn more?

(Other definitions for phaeton that I've seen before include "transport" , "4-wheeled carriage" , "Coach" .)

I've seen this clue in The Globe and Mail.

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