Notation 8.5.9.9. Let $\operatorname{N}_{\leq 2}(\operatorname{Idem})$ denote the simplicial set described in Example 8.5.8.3. By virtue of Proposition 2.4.4.3, we can choose a simplicial category $\operatorname{\mathcal{C}}$ and a morphism of simplicial sets $u: \operatorname{N}_{\leq 2}(\operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a simplicial path category for $\operatorname{N}_{\leq 2}(\operatorname{Idem})$. The morphism $u$ carries the unique vertex $\widetilde{X}$ of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$ to an object $X \in \operatorname{\mathcal{C}}$. Set $E_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet }$, which we regard as a simplicial monoid. Evaluating the morphism $u$ on the nondegenerate edge $\widetilde{e}$ of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$, we obtain a morphism $e: X \rightarrow X$ in the category $\operatorname{\mathcal{C}}$, which we can view as a vertex of the simplicial set $E_{\bullet }$. Evaluating the morphism $u$ on the nondegenerate $2$-simplex of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$, we obtain an edge $h: e^2 \rightarrow e$ in the simplicial monoid $E_{\bullet }$.
Let $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ denote the nerve of the Thompson groupoid. The concatenation functor of Notation 8.5.9.4 endows $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ with the structure of a simplicial monoid. We let $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$ denote the simplicial category given by delooping $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ (Example 2.4.2.3), so that the homotopy coherent nerve of $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$ can be identified with the Duskin nerve of the strict $2$-category $B \mathrm{Dy}$ (see Example 2.4.3.11). It follows that the partial idempotent $\iota $ of Notation 8.5.9.4 can be identified with a morphism from $\operatorname{N}_{\leq 2}( \operatorname{Idem})$ to $\operatorname{N}_{\bullet }^{\operatorname{hc}}( B \operatorname{N}_{\bullet }( \mathrm{Dy} ) )$, which factors unique as a composition
for some simplicial functor $F: \operatorname{\mathcal{C}}\rightarrow B \operatorname{N}_{\bullet }( \mathrm{Dy} )$, which we can identify with a homomorphism of simplicial monoids
By construction, the homomorphism $\varphi $ carries the vertex $e$ to the object $[0,1] \in \mathrm{Dy}$, and the edge $h: e^2 \rightarrow e$ to the dyadic homeomorphism