A question about the dual map of $A \otimes B \to B$.

Question

Let $A$, $B$ be two algebras. Suppose that we have an action $\varphi: A \otimes B \to B$ and there is a pairing $\psi: A \otimes B \to \mathbb{C}$. The action $\varphi$ induces a map $\delta: B \to A^* \otimes B$ defined as follows: $\delta(b) = \sum b_{(1)} \otimes b_{(2)}$, $\sum b_{(1)} \otimes b_{(2)}$ is the notation, $b_{(1)} \in A^*$, $b_{(2)} \in B$, $b_{(1)}(a) b_{(2)} = \varphi(a, b)$, $b_{(1)}(a) = \psi(b_{(1)}^*, a)$. Is the map $\delta$ a homomorphism of algebras? Thank you very much.