  T. Shudo and H. Miyamito \cite{SM78} showed that $C$ can be decomposed into a
direct sum of its indecomposable subcoalgebras of $C$.
  Y.H. Xu \cite {XF92} showed that the decomposition was unique. He also showed
that $M$ can uniquely be decomposed into a direct sum of the weak-closed
indecomposable subcomodules of $M$(we call the decomposition the weak-closed
indecomposable decomposition) in \cite{XSF94}. In this paper, we give the
relation between the two decomposition. We show that if $M$ is a full,
$W$-relational hereditary $C$-comodule, then the following conclusions hold:
(1) $M$ is indecomposable iff $C$ is indecomposable; (2) $M$ is
relative-irreducible iff $C$ is irreducible; (3) $M$ can be decomposed into a
direct sum of its weak-closed relative-irreducible subcomodules iff $C$ can be
decomposed into a direct sum of its irreducible subcoalgebras. We also obtain
the relation between coradical of $C$- comodule $M$ and radical of algebra
$C(M)^*$
