  The mass spectra and wave functions for the doubly heavy baryons are computed
under the picture that the two heavy quarks inside a doubly heavy baryon, such
as two $c$-quarks in $\Xi_{cc}$, combine into a heavy `diquark core' in color
anti-triplet firstly, then the diquark core turns into a color-less doubly
heavy baryon via combining the light $q$-quark inside the baryon. Namely both
of the combinations, the two heavy quarks inside the baryon into a diquark core
in color anti-triplet and the heavy diquark core with the light quark into the
baryon, are depicted by relativistic Bethe-Salpeter equations (BSEs) with an
accordingly QCD inspired kernel respectively, although in the paper only the
heavy diquark cores with the quantum numbers $J^P=1^+$ are considered. Since
the `second combination' is of the heavy diquark core and the light quark, so
the structure effect of the diquark core to the relevant kernel of the BSE is
specially considered in terms of the diquark-core wave functions. The mass
spectra and wave functions for the `low-laying' doubly heavy baryons in the
flavors $(ccq)$, $(bcq)$ and $(bbq)$ and in the quantum numbers
$J^P=\frac{1}{2}^+$, $J^P=\frac{3}{2}^+$, achieved by solving the equations
under the so-called instantaneous approximation, are presented properly and
some comparisons with the others' results under different approaches in the
literature are made.
