University official: 'Everyman's wearing, lower in reputation' (6)
I believe the answer is:
demean
'lower in reputation' is the definition.
'university official everyman's wearing' is the wordplay.
'university official' becomes 'dean' (I've seen this before).
'everyman' becomes 'me' (where 'Everyman' is the setter of the crossword).
'wearing' is an insertion indicator.
'dean' going around 'me' is 'DEMEAN'.
(Other definitions for demean that I've seen before include "Lower in reputation, humiliate" , "Cause to lose respect" , "Lower in status or reputation" , "Discredit" , "Degrade, humiliate" .)