The writer twice revealing an element of culture (4)
I believe the answer is:
meme
'an element of culture' is the definition.
(a meme is an idea spread through culture)
'the writer twice' is the wordplay.
'the writer twice' means two replacements for 'writer'.
'writer' becomes 'me' (the person who wrote the clues).
'writer' becomes 'me' (the person who wrote the clues).
'me'+'me'='MEME'
'revealing' acts as a link.
(Other definitions for meme that I've seen before include "Behavioural trait" , "Question that spreads on the internet" , "Cultural belief that assumes a life of its own" , "Concept that spreads rapidly on the Internet" , "online phenomenon" .)