The most common and most likely answer to this crossword clue is the 6 letter word MATRIC
I've seen this crossword clue in The Times
M
A
T
R
I
C
Buddy explains!

The straight part
We know that "University procedure" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case university procedure = matric.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does MATRIC mean?
MATRIC noun- The final year of high school.
- Someone in their final year of high school.
- Of or pertaining to matrices.