Nip up to fasten (3)
I believe the answer is:
pin
'fasten' is the definition.
(I know this)
'nip up' is the wordplay.
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'nip' back-to-front is 'PIN'.
'to' is the link.
(Other definitions for pin that I've seen before include "Initially, confidential number for use of bank card" , "Small object to fasten or attach" , "Hold (down)" , "Fastener; brooch" , "Debit-card number" .)