  We investigate the Josephson effects in the junction formed by
\emph{DIII}-class topological and $s$-wave superconductors, by embedding a
quantum dot in the junction. Three dot-superconductor coupling manners are
considered, respectively. As a result, the Josephson current is found to
oscillate in $2\pi$ period. Moreover, the presence of Majorana doublet in the
\emph{DIII}-class superconductor renders the current finite at the case of zero
phase difference, with its sign determined by the fermion parity of such a
jucntion. In addition, the dot-superconductor coupling plays a nontrivial role
in adjusting the Josephson current. When the $s$-wave superconductor couples to
the dot in the weak limit, the current direction will have an opportunity to
reverse. The results in this work will be helpful for understanding the
transport properties of the \emph{DIII}-class superconductor.
