Proposition 7.4.5.18. Let $\lambda $ be an uncountable cardinal, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$ be the covariant transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Assume that $\operatorname{\mathcal{C}}$ is $\kappa $-small, where $\kappa = \mathrm{cf}(\lambda )$ is the cofinality of $\lambda $. Let $W$ be the collection of all $U$-cocartesian edges $\operatorname{\mathcal{E}}$. Then the $\infty $-category $\operatorname{\mathcal{E}}[W^{-1}]$ is a colimit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \lambda }$. Moreover, the colimit of $\mathscr {F}$ is preserved by the inclusion functors $\operatorname{\mathcal{QC}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}^{< \mu }$ for $\mu \geq \lambda $.
Proof. By virtue of Proposition 7.4.5.16, there exists a pullback diagram
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 $W$. It follows from Remark 7.4.5.17 that $\overline{U}$ is essentially $\lambda $-small. Using Corollary 5.6.5.13, we can extend $\mathscr {F}$ to a diagram $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$ which is a covariant transport representation for $\lambda $. It follows from Theorem 7.4.5.11 that $\overline{\mathscr {F}}$ is a colimit diagram in $\operatorname{\mathcal{QC}}^{< \lambda }$ (and that it remains a colimit diagram in $\operatorname{\mathcal{QC}}^{< \mu }$ for $\mu \geq \lambda $). $\square$