  One Majorana doublet can be realized at each end of the
time-reversal-invariant Majorana nanowires. We investigate the Josephson effect
in the Majorana-doublet-presented junction modified by different inter-doublet
coupling manners. It is found that when the Majorana doublets couple indirectly
via a non-magnetic quantum dot, only the normal Josephson effects occur, and
the fermion parity in the system just affects the current direction and
amplitude. However, in the odd-parity case, applying finite magnetic field on
the quantum dot can induce the appearance of the fractional Josephson effect.
Next, when the direct and indirect couplings between the Majorana doublets
coexist, no fractional Josephson effect takes place, regardless of finite
magnetic field on the quantum dot. Instead, the $\pi$-period current has an
opportunity to appear in some special cases. All the results are clarified by
analyzing the influence of the fermion occupation in the quantum dot on the
parity conservation in the whole system. We ascertain that this work will be
helpful for describing the dot-assisted Josephson effect between the Majorana
doublets.
