Cautious fighting by Dempsey, ultimately (4)
I believe the answer is:
wary
'cautious' is the definition.
('wary' can be similar in meaning to 'cautious')
'fighting by dempsey ultimately' is the wordplay.
'fighting' becomes 'war'.
'by' is a charade indicator (letters next to each other).
'ultimately' indicates one should take the final letters.
The last letter of 'dempsey' is 'y'.
'war'+'y'='WARY'
(Other definitions for wary that I've seen before include "Displaying caution" , "Cautious and vigilant.17 Add up for infant" , "Cautious - vigilant" , "Cautious of danger" , "Circumspect" .)