  We discuss the change of the Kondo effect in the Josephson junction formed by
the indirect coupling between a one-dimensional \emph{DIII}-class topological
and s-wave superconductors via a quantum dot. By performing the
Schrieffer-Wolff transformation, we find that the single-electron occupation in
the quantum dot induces various correlation modes, such as the Kondo and
singlet-triplet correlations between the quantum dot and the $s$-wave
superconductor and the spin exchange correlation between the dot and Majorana
doublet. Moreover, it plays a nontrivial role in modifying the Josephson
effect, leading to the occurrence of anisotropic and high-order Kondo
correlation. In addition, due to the quantum dot in the Kondo regime, extra
spin exchange correlations contribute to the Josephson effect as well.
Nevertheless, if the \emph{DIII}-class topological superconductor degenerates
into \emph{D}-class because of the destruction of time-reversal invariance, all
such terms will disappear completely. We believe that this work shows the
fundamental difference between the \emph{D}- and \emph{DIII}-class topological
superconductors.
