Kerodon

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Corollary 8.5.1.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the inclusion map $\mathcal{R} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$ induces a trivial Kan fibration

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \{ 0 < 2 \} ), \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]

In particular, every retraction diagram in $\operatorname{\mathcal{C}}$ can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$, which is uniquely determined up to isomorphism.