Runs off after snatching coat, to be precise (5,5)
I believe the answer is:
split hairs
'be precise' is the definition.
(splitting hairs means being pedantic)
'runs off after snatching coat' is the wordplay.
'runs off' becomes 'splits' (split can mean to run away).
'after snatching' indicates putting letters inside (inserted letters are snatched).
'coat' becomes 'hair' (an animal's coat).
'splits' placed around 'hair' is 'SPLIT HAIRS'.
'to' acts as a link.
(Other definitions for split hairs that I've seen before include "Quibble" , "salon complaint?" , "Make petty distinctions" , "a trichologist may deal with" , "cavil exceedingly" .)