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Lemma 5.6.9.2. Let $\operatorname{\mathcal{E}}$ be an essentially small $\infty $-category. Then the simplicial set $\operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 )$ is a contractible Kan complex.

Proof. It follows from Lemma 5.6.8.7 that the simplicial set $\operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 )$ is a Kan complex. Since $\operatorname{\mathcal{E}}$ is essentially small, the Kan complex $\operatorname{TW}(\operatorname{\mathcal{E}}/ \Delta ^0)$ is nonempty. It will therefore suffice to show that the diagonal map

\[ \delta : \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 ) \rightarrow \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 ) \times \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 ) \]

is a homotopy equivalence (Corollary 3.2.7.6). Unwinding the definitions, we see that $\delta $ factors as a composition

\begin{eqnarray*} \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 ) & \xrightarrow {\delta '} & \operatorname{Fun}( \Delta ^1, \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0) ) \\ & \simeq & \operatorname{TW}( \Delta ^1 \times \operatorname{\mathcal{E}}/ \Delta ^1 ) \\ & \xrightarrow {\delta ''} & \operatorname{TW}( \operatorname{\partial \Delta }^1 \times \operatorname{\mathcal{E}}/ \operatorname{\partial \Delta }^1 ) \\ & \simeq & \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^0 ) \times \operatorname{TW}(\operatorname{\mathcal{E}}/ \Delta ^0 ). \end{eqnarray*}

Since the $1$-simplex $\Delta ^1$ is contractible (Example 3.2.6.3), the morphism $\delta '$ is a homotopy equivalence. It will therefore suffice to show that the restriction map $\delta ''$ is a homotopy equivalence, which follows from Lemma 5.6.9.1. $\square$