  The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [7] as a
generalization of Hope group caolgebra, introduced by Turaev in 2000 [5], in
the non-unital case. We prove that the concepts introduced by A.Van Daele in
constructing multiplier Hopf algebra \cite{4} can be adapted to serve again in
our construction. A multiplier Hopf group coalgebra is a family of algebras
$A=\{A_{\alpha}\}_{\alpha \in \pi}$, ($\pi$ is a discrete group) equipped with
a family of homomorphisms
$\Delta=\{\Delta_{\alpha,\beta}:A_{\alpha\beta}\longrightarrow
M(A_{\alpha}\otimes A_{\beta})\}_{\alpha,\beta \in \pi}$ which is called a
comultiplication under some conditions, where $M(A_{\alpha}\otimes A_{\beta})$
is the multiplier algebra of $A_{\alpha}\otimes A_{\beta}$.
  In 2003 A. Van Daele suggest a new approach to study the same structure by
consider the direct sum of the algebras $A_p$'s which will be a multiplier Hopf
algebra called later group cograded multiplier Hope algebra \cite{11}. And
hence there exist a one to one correspondence between multiplier Hopf Group
Coalgebra and group cograded multiplier Hopf algebra. By using this one-one
correspondence we studied multiplier Hopf Group Coalgebra \\
