write up
write up {v.}
1. To write the story of; describe in writing; give a full account of.
Reporters from many newspapers are here to write up the game.
The magazine is writing up the life of the President.
2. To put something thought or talked about into writing; finish writing (something).
John took notes of what the teacher said in class and he wrote them up when he got home.
The author had an idea for a story when he saw the old house, and he wrote it up later.
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Source: A Dictionary of American Idioms