The most common and most likely answer to this crossword clue is the 10 letter word COMMISSION
I've seen this crossword clue in The Guardian
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More clues leading to the result COMMISSION
Buddy explains!

The straight part
We know that "Appoint" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case appoint = commission.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 COMMISSION mean?
COMMISSION noun- A sending or mission (to do or accomplish something).
- An official charge or authority to do something, often used of military officers.
- The thing to be done as agent for another.
- A body or group of people, officially tasked with carrying out a particular function.
- A fee charged by an agent or broker for carrying out a transaction.
- The act of committing (e.g. a crime).
- To send or officially charge someone or some group to do something.
- To place an order for (often piece of art)
- To put into active service