Objection raised after City's home fixture (7)
I believe the answer is:
bathtub
'home fixture' is the definition.
(fixture found in the home)
'objection raised after city's' is the wordplay.
'objection' becomes 'but' (as in 'no ifs, no buts').
'raised' is a reversal indicator (going up in a down clue).
'after' means one lot of letters go next to another.
'city' becomes 'bath' (English city).
'but' reversed gives 'tub'.
'tub' put after 'bath' is 'BATHTUB'.
(Other definitions for bathtub that I've seen before include "Object used for washing bodies" , "Archimedes' reputedly was here before his 'Eureka'" , "Where to soak (mainly US)" , "Container for the body during ablutions" , "Washing vessel" .)