Supporter at home showing intelligence (5)
I believe the answer is:
brain
'intelligence' is the definition.
(brain is a kind of intelligence)
'supporter at home' is the wordplay.
'supporter' becomes 'bra' (a bra gives support to breasts).
'at home' becomes 'in' (similar in meaning).
'bra'+'in'='BRAIN'
'showing' is the link.
(Other definitions for brain that I've seen before include "Haemorrhaging of talent" , "Thinking organ" , "The grey matter" , "Thinking thing" , "It controls the central nervous system" .)