a unital $C^*$-algebra induced by a left coaction {#sec:coaction}
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We use the same notations as in the previous section and also we suppose that the same assumptions as in the previous section. We note that $C=A\rtimes_{\rho, u}H$ and $D=B\rtimes_{\sigma, v}H$. We also note that $D$ is anti-isomorphic to ${}_C {\mathbf B}(Y)$, the $C^*$-algebra of all adjointable left $C$-modules maps on $Y$. We identify $D$ with ${}_C {\mathbf B}(Y)$. Hence for any $T, R\in {}_C {\mathbf B}(Y)$ we can write their product as follows: For any $y\in Y$, $$(TR)(y)=R(T(y)) .$$ For any $y, z\in Y$, let $\Theta_{y, z}$ be the rank-one left $C$-module map on $Y$ defined by $$\Theta_{y, z}(x)={}_C {\langle}x, z {\rangle}\cdot y$$ for any $x\in Y$. Then since $Y$ is of finite type in the sense of Kajiwara and Watatani [@KW1:bimodule], ${}_C {\mathbf B}(Y)$ is the linear span of the above rank-one left $C$-module maps on $Y$ by [@KW1:bimodule]. We define a bilinear map $``\rightharpoonup"$ from $H^0 \times D$ to $D$ so that $$\psi\rightharpoonup{\langle}y, z {\rangle}_D ={\langle}S^0 (\psi_{(1)}^* )\triangleright y \, , \, \psi_{(2)}\triangleright z {\rangle}_D$$ for any $\psi \in H^0$ and $y, z\in Y$. Hence since ${\langle}y, z {\rangle}_D =\Theta_{z, y}$ for any $y, z\in Y$, we 