\] Let $(G, P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. A homomorphism $\pi:B\to {\operatorname{\mathcal{NO}}_{\!X}}^{r}$ is injective if and only if there is a normal coaction $\beta$ of $G$ on $B$ such that $\pi$ is $\beta-\nu^n$ equivariant and $\pi\vert_{B_e}$ is injective.  Apply Corollary \[cor: