A snob set up institute for Japanese art (6)
I believe the answer is:
bonsai
'japanese art' is the definition.
(I've seen this before)
'a snob set up institute' is the wordplay.
'set up' says the letters should be written backwards (in a down clue, letters go up).
'institute' becomes 'i' (common abbreviation as in MIT).
'a'+'snob'='asnob'
'asnob' in reverse letter order is 'bonsa'.
'bonsa'+'i'='BONSAI'
'for' acts as a link.
(Other definitions for bonsai that I've seen before include "Tree dwarfing art" , "Japanese art of growing miniature trees" , "Japanese method of cultivating miniature trees" , "Mini-tree growing" , "Dwarf trees and shrubs" .)