# Kerodon

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Corollary 7.4.3.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between small simplicial sets, and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a covariant transport representation of $U$. Then the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ admits a colimit in $\operatorname{\mathcal{QC}}$. Moreover, an object $\operatorname{\mathcal{D}}\in \operatorname{\mathcal{QC}}$ is a colimit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if it is equivalent to the localization $\operatorname{\mathcal{E}}[W^{-1}]$, where $W$ is the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

Proof. Let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. By virtue of Proposition 7.4.3.9 (and Remark 7.4.3.10), there exists a pullback diagram of small simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } }$

where $\overline{U}$ is a cocartesian fibration, and a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ which exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. Applying Corollary 5.6.5.11, we see that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ extends to a covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$. By virtue of Theorem 7.4.3.6, this extension is a colimit diagram carrying ${\bf 0}$ to the $\infty$-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } \simeq \operatorname{\mathcal{E}}[W^{-1}]$. $\square$