If you consider the category of $G$-graded vector spaces, the homogenous elements can be written as $U_g$ for $g \in G$.
The category of $G$-graded vector spaces is isomorphic to the category of $k^G-$modules (where $k^G$ is the dual Hopf algebra to the group algebra $k[G]$).
I have a representation over $G$: $$\rho_0: G \rightarrow \mathrm{End}(V_0)$$ and want to define the following map for $V, V_0$ vector spaces and $U_g$ a graded vector spaces: $$ \lambda_{g, V}: V_0 \otimes (V \otimes U_g) \rightarrow (V_0 \otimes V) \otimes U_g, v_0 \otimes v \otimes u \mapsto \rho_0(g)(v_0) \otimes v \otimes u$$
As one sees, the $g$ which I use in the representation is "taken from" the grading of the $G$-graded homogenous vector space.
Now consider the more general case: Let $H$ be a finite dimensional Hopf algebra and look at the modules over the dual:
$$H^*-\mathrm{mod}$$
I want to adapt a proof - in which I need the above map $\lambda_{g, V}$ - to this new, more general situation.
Here $U_g$ would be replaced by a module $N \in H^*-\mathrm{mod}$, I still have a representation $\rho_0: H \rightarrow \mathrm{End}(V_0)$, but now I don't know which $h$ I have to use. I guess what I am asking is:
Are there any components in $H^*-\mathrm{mod}$ which can be identified with an element in $H$? I guess, homogenous doesn't make sense as I don't have a grading, but kind of "basic" elements.
We have that $H^* \subset H^*-\mathrm{mod}$, but the elements in $H^*$ are maps from $H$ to $k$ - still now indexing with an $h \in H$ possible.
I'm not so good at writing questions, so I you have any questions or comments or if I have missed something, please comment!
A $G$-graded vector space is a comodule over the Hopf algebra $\Bbbk[G]$. Graded components are, broadly speaking, suitable subcomodules of your $G$-graded vector space: those for which the coaction $\delta:U\to U\otimes \Bbbk[G]$ behaves as follows: $$\delta: U_g\to U_g\otimes \Bbbk[G]: u\mapsto u\otimes g.$$ Therefore, if you want to generalize your construction you may perform something similar to the following. Let $H$ be any Hopf algebra. Take $V_0$ to be a left $H$-module, $V$ a vector space, $U$ a right $H$-comodule and define $$ \lambda: V_0\otimes (V\otimes U) \to (V_0\otimes V)\otimes U: v_0\otimes v\otimes u\mapsto \sum_uu_{[1]}v_0 \otimes v\otimes u_{[0]}, $$ where $\delta(u)=\sum_u u_{[0]} \otimes u_{[1]}$ by resorting to Sweedler's Sigma Notation.
However, if you may provide some more context concerning your question (such as why you put the parenthesis in that particular position, why you would like to define such a $\lambda$, etc.) maybe we can be more helpful.