  As endpoints of the hierarchical mass-assembly process, the stellar
populations of local early-type galaxies encode the assembly history of
galaxies over cosmic time. We use Horizon-AGN, a cosmological hydrodynamical
simulation, to study the merger histories of local early-type galaxies and
track how the morphological mix of their progenitors evolves over time. We
provide a framework for alleviating `progenitor bias' -- the bias that occurs
if one uses only early-type galaxies to study the progenitor population.
Early-types attain their final morphology at relatively early epochs -- by
$z\sim1$, around 60 per cent of today's early-types have had their last
significant merger. At all redshifts, the majority of mergers have one
late-type progenitor, with late-late mergers dominating at $z>1.5$ and
early-early mergers becoming significant only at $z<0.5$. Progenitor bias is
severe at all but the lowest redshifts -- e.g. at $z\sim0.6$, less than 50 per
cent of the stellar mass in today's early-types is actually in progenitors with
early-type morphology, while, at $z\sim2$, studying only early-types misses
almost all (80 per cent) of the stellar mass that eventually ends up in local
early-type systems. At high redshift, almost all massive late-type galaxies,
regardless of their local environment or star-formation rate, are progenitors
of local early-type galaxies, as are lower-mass (M$_\star$ $<$ 10$^{10.5}$
M$_{\odot}$) late-types as long as they reside in high density environments. In
this new era of large observational surveys (e.g. LSST, JWST), this study
provides a framework for studying how today's early-type galaxies have been
built up over cosmic time.
