# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Notation 8.5.6.15. Let $\operatorname{Ret}$ denote the category of Construction 8.5.0.1. 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 $\widetilde{Y}$.