  We critically examine the role that correlations established between a system
and fragments of its environment play in characterising the ensuing dynamics.
We employ a class of dephasing models where the state of the initial
environment represents a tunable degree of freedom that qualitatively and
quantitatively affects the correlation profiles, but nevertheless results in
the same reduced dynamics for the system. We apply recently developed tools for
the characterisation of non-Markovianity to carefully assess the role that
correlations, as quantified by the (quantum) Jensen-Shannon divergence and
relative entropy, as well as changes in the environmental state, play in
whether the conditions for classical objectivity within the quantum Darwinism
paradigm are met. We demonstrate that for precisely the same non-Markovian
reduced dynamics of the system arising from different microscopic models, some
will exhibit quantum Darwinistic features, while others show no meaningful
notion of classical objectivity is present. Furthermore, our results highlight
that the non-Markovian nature of an environment does not a priori prevent a
system from redundantly proliferating relevant information, but rather it is
the system's ability to establish the requisite correlations that is the
crucial factor in the manifestation of classical objectivity.
