Keep an eye on the time (5)
I believe the answer is:
This clue is a double definition.
'keep an eye on' is the first definition.
'time' is the second definition.
'the' acts as a link.
(Other definitions for watch that I've seen before include "Vigil" , "Look at attentively" , "Timepiece - look out" , "period of duty" , "Observe - guard" .)