Let $G', G$ be groups and we have a left action $G' \times G \to G$. Which of the following is the dual of $G' \times G \to G$?
(1) $\mathbb{C}[G] \to \mathbb{C}[G'] \otimes \mathbb{C}[G]$, $f \mapsto \sum f_{(-1)} \otimes f_{(0)}$, $f(h.a)=f_{(-1)}(h)f_{(0)}(a)$, $h \in G'$, $a \in G$.
(2) $\mathbb{C}[G] \to \mathbb{C}[G] \otimes \mathbb{C}[G']$, $f \mapsto \sum f_{(0)} \otimes f_{(-1)}$, $f(h.a)=f_{(-1)}(h)f_{(0)}(a)$, $h \in G'$, $a \in G$.
Thank you very much.
I'm slightly unclear on your notation, but as remarked in the comments, the ($\mathbb C$-linear) dual of $$\alpha: (\mathbb C[G'] \otimes \mathbb C[G]) \to \mathbb C[G]$$ is $$\alpha^*: \mathbb C[G]^* \to (\mathbb C[G'] \otimes \mathbb C[G])^*$$ that takes $f \in \mathbb C[G]^*$ to $\alpha^*(f)(g' \otimes g) = f(g' \cdot g)$.
If we write $g^* \in \mathbb C[G]^*$ for the element dual to $g \in \mathbb C[G]$, then we see that $\alpha^*(g^*)(h' \otimes h) = g^*(h' \cdot h)$ so $$\alpha^*(g^*) = \sum_{h', h \text{ where } h' \cdot h = g} (h' \otimes h)^*$$
You can then identify $\mathbb C[G]$ and $\mathbb C[G'] \otimes \mathbb C[G]$ with their respective duals.