Kerodon

Remark 7.4.4.4. Using Theorem 7.4.3.6, one can show that conclusion of Lemma 7.4.4.3 holds more generally under the assumption that $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ is a left cofinal morphism of simplicial sets. For simplicity, let us assume that each of the simplicial sets appearing in the statement of Lemma 7.4.4.3 is small. Using Proposition 7.4.3.9, we can assume that $U$ is the pullback of a cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ for which the covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ exhibits the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. Using Theorem 7.4.3.6, we deduce that the covariant transport representation $\operatorname{Tr}= \operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$ is a colimit diagram. Since $F$ is right cofinal, it follows that the restriction $\operatorname{Tr}|_{ \operatorname{\mathcal{C}}_0^{\triangleright } }$ is also a colimit diagram (Corollary 7.2.2.3). Applying Theorem 7.4.3.6 again, we conclude that $\mathrm{Rf}|_{\operatorname{\mathcal{E}}_0}$ exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}_0$ with respect to $W_0$, so that $\widetilde{F}$ induces an equivalence $\operatorname{\mathcal{E}}_0[W_0^{-1}] \xrightarrow {\sim } \operatorname{\mathcal{E}}[ W^{-1} ]$.