Let $(S,\Delta)$ be a compact quantum group and $U\in M(K(H)\otimes S)$ be a unitary corepresentation of $S$ on $H$. Let $\phi $ be a state of $S$. Let $A$ be a sub-$C^*$-algebra of $B(H)$. If $a\in A$, we let $Ad_U(a)=U(a\otimes 1)U^*$.
Why the element $(Id\otimes \phi)\circ Ad_U(a)$ is well-defined?
I don't understand the action of $Id\otimes \phi$ on $Ad_U(a)$.
Remark: I try to understand Definition 2.1 (page 3) of