Kerodon

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Example 7.4.5.7. Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category. Applying Example 7.4.5.6 in the special case $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_1$, we deduce that $\operatorname{\mathcal{C}}$ can be realized as the colimit of a diagram

\[ \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}\quad \quad C \mapsto \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} , \]

given by covariant transport for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.