Winner, Scotsman, becoming strait-laced (9)
I believe the answer is:
victorian
'strait-laced' is the definition.
('victorian' can be a synonym of 'strait-laced')
'winner scotsman' is the wordplay.
'winner' becomes 'victor' ('victor' can be a synonym of 'winner').
'scotsman' becomes 'ian'.
'victor'+'ian'='VICTORIAN'
'becoming' is the link.
(Other definitions for victorian that I've seen before include "from Ballarat, perhaps" , "Of the age of a long-lived British queen" , "in part of Australia" , "Morally strict" , "from somewhere in Oz" .)