  We characterize the condition for two finite index endomorphisms on an AFD
factor to be approximately unitarily equivalent. The characterization is given
by using the canonical extension of endomorphisms, which is introduced by
Izumi. Our result is a generalization of the characterization of approximate
innerness of endomorphisms of the AFD factors, obtained by
Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not
depend on the types of factors, is based on recent development on the Rohlin
property of flows on von Neumann algebras.
