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Proposition The morphism $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ of Notation is both left and right cofinal.

Proof. We will show that $Q$ is left cofinal; a similar argument will show that it is right cofinal. By virtue of Theorem, it will suffice to show that the simplicial set $K = \operatorname{Spine}[\operatorname{\mathbf{Z}}] \times _{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } \operatorname{N}_{\bullet }( \operatorname{Idem})_{/ \widetilde{X}}$ is weakly contractible. Let us identify the vertices of $K$ with pairs $(n,f)$, where $n$ is an integer and $f: \widetilde{X} \rightarrow \widetilde{X}$ is a morphism in the category $\operatorname{Idem}$. Unwinding the definitions, we see that $K$ is the $1$-dimensional simplicial set associated to the direct graph given in the diagram

\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [dr] \ar [r] & (-1, \widetilde{e}) \ar [dr] \ar [r] & (0, \widetilde{e}) \ar [dr] \ar [r] & (1, \widetilde{e}) \ar [dr] \ar [r] & \cdots \\ \cdots & (-1,\operatorname{id}_{\widetilde{X}} ) & (0, \operatorname{id}_{\widetilde{X}} ) & (1, \operatorname{id}_{\widetilde{X}} ) & \cdots . } \]

The inclusion of the upper part of the diagram determines a monomorphism of simplicial sets $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow K$ which is left anodyne (since it is a pushout of a coproduct of countably many copies of the inclusion map $\{ 0\} \hookrightarrow \Delta ^1$), and therefore a weak homotopy equivalence (Proposition The desired result now follows from the weak contractibility of the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ (Remark $\square$