Let $H$ ba a Hopf algebra with coaction $\Delta: H \to H \otimes H$. Denote the action of $\Delta$ by $\Delta (h) = h_{(1)} \otimes h_{(2)}$. I was wondering if every element of $H$ can arise as a $h_{(2)}$, for some $h$. To be more precise, for any $g \in H$, does there exist a $h \in H$, such that $\Delta(h) = f \otimes g + \sum_i h_i \otimes h'_i$, for some set $h'_i$ which is linearily indpt with $g$? Or, alternatively, does there exist $H' \subseteq H$, such that $\Delta(H) \subseteq H \otimes H'$?
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Counitarity implies that $$(\varepsilon\otimes\mathrm{id}_H\circ\Delta)(H)=H.$$ If $\Delta(H)\subseteq H\otimes H'$ for some subspace $H'\subseteq H$, then this tells us since $\varepsilon$ takes scalar values! that $H\subseteq H'$. 

