Consider a Hopf algebra $H$ where $S$ is the antipode and $\Delta$ is the coproduct. Given $a \in H$, I want to know if there always exists an element $b \in H$ satisfying the 'flip' property: $$\Delta (b) (1 \otimes a) = Sa \otimes 1$$
I know that this property is satisfied for quantum double of $G$, which can be seen as follows. Let $\{k\}_{k \in G}$ be a basis for $\mathbb{C}(G)$ (group algebra) and let $\{g^*\}_{g \in G}$ be a basis for $\mathcal{F}(G)$ (space of complex valued functions over $G$). Then for any element $a = k g^*$, the element $b = \sum_{l \in G} k l^*$ does the job.
But is it a universal property of Hopf algebras? How can I see it if so?