Written by yours truly, second note (4)
I believe the answer is:
memo
'note' is the definition.
(a memo is a short note)
'written by yours truly second' is the wordplay.
'written by' says to put letters next to each other.
'yours truly' becomes 'me' (the person who set this crossword).
'second' becomes 'mo' (both are short periods of time).
'me'+'mo' is 'MEMO'.
(Other definitions for memo that I've seen before include "Office note" , "It will remind" , "Brief note, usually between co-workers" , "diplomatic communication?" , "A brief record of events" .)