The most common and most likely answer to this crossword clue is the 4 plus 2 plus 2 letter word FELO DE SE
I've seen this crossword clue in The Guardian
F
E
L
O
D
E
S
E
Buddy explains!
The straight part
We know that "putting oneself down" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case putting oneself down = felo de se.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 FELO DE SE mean?
FELO DE SE noun- One who deliberately puts an end to his own existence, or loses his life while engaged in the commission of an unlawful or malicious act; a suicide.