.  Let $(G,P)$ be a quasi-lattice ordered group and $X$ a compactly aligned product system over $P$ of $C^*$-correspondences over $A$. A surjective homomorphism $\pi:{\operatorname{\mathcal{NT}}}(X)\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NT}}}(X)^\delta$ and there is a 