Conceited son supported by fool (4)
I believe the answer is:
smug
'conceited' is the definition.
(thesaurus)
'son supported by fool' is the wordplay.
'son' becomes 's'.
'supported by' says to put letters next to each other (letters go below - in down clue).
'fool' becomes 'mug' ('mug' can be a synonym of 'fool'**).
's'+'mug'='SMUG'
(Other definitions for smug that I've seen before include "Unpleasantly complacent" , "Irritatingly self-satisfied" , "goody-goody" , "Self-satisfied about gums" , "Full of oneself" .)