Objections rejected in the end (4)
I believe the answer is:
stub
'the end' is the definition.
(eg the stub of a cigarette)
'objections rejected' is the wordplay.
'objections' becomes 'buts' (as in 'no ifs, no buts').
'rejected' is a reversal indicator.
'buts' written backwards gives 'STUB'.
'in' is the link.
(Other definitions for stub that I've seen before include "End of cigarette or counterfoil of cheque" , "evidence of smoking, for example" , "Bust (anag.)" , "Put out" , "Small piece, especially of something cut down or worn away" .)