  Production of the doubly heavy-flavored hadrons ($B_c$ meson, doubly heavy
baryons $\Xi_{cc}$, $\Xi_{bc}$, $\Xi_{bb}$, their excited states and
antiparticles of them as well) at $e^+e^-$ colliders is investigated under two
different approaches: $LO$ (leading order QCD complete calculation) and $LL$
(leading logarithm fragmentation calculation). The results for the production
obtained by the approaches $LO$ and $LL$, including the angle distributions of
the produced hadrons with unpolarized and polarized incoming beams, the
behaviors on the energy fraction of the produced doubly heavy hadron and
comparisons between the two approaches' results, are presented in terms of
tables and figures. Thus characteristics of the production and uncertainties of
the approaches are shown precisely, and it is concluded that only if the
colliders run at the eneries around $Z$-pole (which may be called as
$Z$-factories) and additionally the luminosity of the colliders is as high as
possible then to study the doubly heavy hadrons thoroughly is accessible.
