  We present a complete topological characterization of a bilayer composite of
two Chern insulators (specifically, Haldane models) and explicitly establish
the bulk-boundary correspondences. We show that an appropriately defined Chern
number accurately maps out all the possible phases of the system and remains
well-defined even in the presence of degeneracies in the occupied bands.
Importantly, our result paves the way for realizing adiabatic preparation of
monolayer Chern insulators. This has been a major challenge till date, given
the impossibility of unitarily connecting inequivalent topological phases. We
show that this difficulty can be circumvented by adiabatically varying the
interlayer coupling in such a way that the system remains gapped at all times.
In particular, a complete knowledge of the phase diagram of the bilayer
composite immediately allows one to identify all such adiabatic passages which
may connect the different Chern inequivalent phases of the individual
monolayers.
