Example 5.6.8.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$ denote the projection map. Note that projection onto the first factor determines a morphism of simplicial sets
\[ \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^{0} ) \rightarrow \operatorname{Fun}( \Delta ^0, \operatorname{\mathcal{QC}}) = \operatorname{\mathcal{QC}}. \]
Unwinding the definitions, we see that the fiber of this morphism over a small $\infty $-category $\operatorname{\mathcal{E}}'$ can be identified with the full subcategory
\[ \operatorname{Equiv}( \operatorname{\mathcal{E}}, \{ \operatorname{\mathcal{E}}' \} \times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{E}}, \{ \operatorname{\mathcal{E}}' \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}})^{\simeq } \]
spanned by the equivalences of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \{ \operatorname{\mathcal{E}}' \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.