Dispatch book left behind by somebody (8)
I believe the answer is:
celerity
'dispatch' is the definition.
(I've seen this before)
'book left behind by somebody' is the wordplay.
'book' becomes 'b' (abbreviation for book).
'left behind by' is a deletion indicator.
'somebody' becomes 'celebrity' (a celebrity might be called a somebody).
'celebrity' with 'b' removed is 'CELERITY'.
(Other definitions for celerity that I've seen before include "Rapidity, speediness" , "showing a good turn of speed" , "Pace" , "Haste" , "Quickness" .)