Poet that's written book on composer (7,6)
I believe the answer is:
matthew arnold
'poet' is the definition.
(19th-century poet)
'written book on composer' is the wordplay.
'written book' becomes 'Matthew' (book of the Bible).
'on' says to put letters next to each other.
'composer' becomes 'Arnold' (composer Malcolm Arnold).
'matthew'+'arnold'='MATTHEW ARNOLD'
'that's' acts as a link.
(Other definitions for matthew arnold that I've seen before include "poet - with 5a" , "Victorian poet, headmaster's son" .)