Rest of university engaged in lie (7)
I believe the answer is:
residue
'rest' is the definition.
('residue' can be a synonym of 'rest')
'university engaged in lie' is the wordplay.
'university' becomes 'u'.
'engaged in' indicates putting letters inside.
'lie' becomes 'reside' (I've seen this in other clues).
'u' inserted into 'reside' is 'RESIDUE'.
'of' is the link.
(Other definitions for residue that I've seen before include "Remains to be seen" , "Whatever is left over" , "Something left over" , "What remains of something" , "What's left" .)