The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

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Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much.