  We consider multi-component dark matter in a model where one dark matter
component is feebly interacting (FIMP) while the second is weakly interacting
(WIMP). The model contains an inert scalar doublet and a complex scalar singlet
and features a discrete $Z_4$ symmetry. We determine the parameter space that
satisfies theoretical constraints, collider constraints, relic density as well
as direct and indirect detection limits. We discuss the possibility to probe
the model through collider and astrophysical searches. We find that it is
possible that the FIMP forms the dominant DM component, in this case the
astrophysical signatures of the WIMP are much suppressed. We also explore the
case where the doublet can decay into pairs of singlets leaving only one DM
component, the FIMP. These scenarios are constrained by BBN and are best
explored at colliders since astrophysical signatures are suppressed.
