Notation 8.5.6.15. Let $\operatorname{Ret}$ denote the category of Construction 8.5.0.2. Then the object $\widetilde{Y} \in \operatorname{Ret}$ is both initial and final. It follows that the diagram $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }(\operatorname{Idem})$ of Proposition 8.5.4.16 admits unique extensions
\[ Q^{-}: \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleleft } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Ret}) \quad \quad Q^{+}: \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Ret}) \]
which carry the cone points to the object $\widetilde{Y}$.