Folding elegantly, banking system backed mate in Paris (7)
I believe the answer is:
origami
'folding elegantly' is the definition.
'origami' can be an answer for 'folding' (origami involves folding). I am unsure of the 'elegantly' bit.
'banking system backed mate in paris' is the wordplay.
'banking system' becomes 'giro' (I've seen this before).
'backed' shows that the letters should be reversed in order.
'mate in paris' becomes 'ami' ('friend' in French).
'giro' in reverse letter order is 'orig'.
'orig'+'ami'='ORIGAMI'
Can you help me to learn more?
(Other definitions for origami that I've seen before include "The folding of sheets" , "that's folding" , "folding stuff?" , "Japanese art form -- I'm a giro (rev)" , "Making of paper shape" .)