The most common and most likely answer to this crossword clue is the 10 letter word COLLEGIATE
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More clues leading to the result COLLEGIATE
Buddy explains!

The straight part
We know that "of educational establishment" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of educational establishment = collegiate.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 COLLEGIATE mean?
COLLEGIATE noun- Another name for a high school (e.g. some high schools are called collegiates rather than high schools)
- A member of a college, a collegian; someone who has received a college education.
- A fellow-collegian; a colleague.
- An inmate of a prison.
- Of, or relating to a college, or college students.
- Collegial.