# Kerodon

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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 $\operatorname{\mathcal{E}}\rightarrow \{ \operatorname{\mathcal{E}}' \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.