  In a previous paper we introduced the concept of semiseparable functor. Here
we continue our study of these functors in connection with idempotent (Cauchy)
completion. To this aim, we introduce and investigate the notions of
(co)reflection and bireflection up to retracts. We show that the (co)comparison
functor attached to an adjunction whose associated (co)monad is separable is a
coreflection (reflection) up to retracts. This fact allows us to prove that a
right (left) adjoint functor is semiseparable if and only if the associated
(co)monad is separable and the (co)comparison functor is a bireflection up to
retracts, extending a characterization pursued by X.-W. Chen in the separable
case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in
the framework of pre-triangulated categories.
