I have a group algebra of a finite group $G$ over $\mathbb{C}$ and $Fun(\mathbb{C}[G])$ represents the linear functions on $\mathbb{C}[G]$. $\mathbb{C}[G] \otimes \mathbb{C}[G]$ denotes the tensor product of $\mathbb{C}[G]$ as vector spaces with multiplication $(g_1 \otimes g_2)*(g_3 \otimes g_4) = (g_1g_3 \otimes g_2g_4)$.
Assume that we have the comultiplication $g \mapsto g \otimes g$, counit $g \mapsto 1$ and antipode $g \mapsto g^{-1}$. These operations make $\mathbb{C}[G]$ a Hopf algebra.
I want to show that algebras $\mathbb{C}[G]$ and $Fun(\mathbb{C}[G])$ are dual to each other by bilinear form $<f,g>=f(g),\ f \in Fun(\mathbb{C}[G]),\ g \in \mathbb{C}[G]$;
But for that I need to represent the image of comultiplication of $f \in Fun(\mathbb{C}[G])$:
$f(g_1,g_2) \in Fun(\mathbb{C}[G] \otimes \mathbb{C}[G])$, (which is $f(g_1*g_2)$), as $X(g_1) \otimes Y(g_2) \in Fun(\mathbb{C}[G]) \otimes Fun(\mathbb{C}[G])$,
but have no idea how to do that.