Let $G$ be a group with $1$ be the identity element of $G$. Let $(C,\bigtriangleup,\epsilon)$ be a coalgebra over a field $K$. $C$ is called $G$-graded if $C$ admits a decomposition as a direct sum of $K$-vector spaces $C=\bigoplus _{\sigma \in G} C_{\sigma}$ such that:
(i) $\bigtriangleup (C_{\sigma}) \subseteq \sum_{\lambda \mu = \sigma}C_{\lambda} \otimes C_{\mu}$ for any $\sigma \in G$.
(ii) $\epsilon (C_{\sigma})=0$ for any $\sigma \neq 1$
I want to prove the following:
Let $M$ be a right C-comodule. Let $F(M)=\bigoplus_{\sigma \in G}M^{\sigma}$ where $M^{\sigma}=M$ for all $\sigma \in G$. We define $\rho_{F(M)}:F(M) \rightarrow F(M) \otimes C$ as follows:
It is enough to define the above map for elements of the form $x=(...,0,m^{\sigma},0,...)$ where $m^{\sigma}=m \in M$ and the other components of $x$ are zeros. Since $\rho_{M}:M \rightarrow M \otimes C\cong \bigoplus_{\lambda \in G}(M \otimes C_{\lambda})$ then $\rho_{M}(m)=\sum_{(m)} m_{0} \otimes m_{1}^{\lambda}$ where $m_{0} \in M$ and $m_{1}^{\lambda} \in C_{\lambda}$
All the above sums are finite.
Now we define $\rho_{F(M)}(x)=\sum_{\tau \in G}(...,m_{0} \otimes m_{1}^{\tau},...)$ where $m_{0}$ is considered as an element in $M^{\sigma \tau^{-1}}$
How to prove that $F(M)$ is a right $C$-comodule with the map $\rho_{F(M)}$.
Note: The above are some statements from a research paper that I am trying to understand.
Small example to clarify things:
Let $G=\{1,a\}$ any group of order two. Assume that $\rho_{M}(m)=m_{0} \otimes c_{0}^{1}+ m_{1} \otimes c_{1}^{a}$ where $m_{i} \in M$, $c_{0}^{1} \in C_{1}$ and $c_{1}^{a} \in C_{a}$
Then $F(M)=M_{1} \oplus M_{a}$ where $M_{1}=M_{a}=M$
So, for example $\rho_{F(M)}(m,0)=(m_{0},0)\otimes c_{0}^{1}+(0,m_{1})\otimes c_{1}^{a}$