Remark 7.4.1.17. In the formulation of Proposition 7.4.1.16, the essential smallness assumption on $\overline{U}$ is not important. If $\overline{U}$ is essentially $\kappa $-small (for some uncountable cardinal $\kappa $ which is not necessary small), then it admits a covariant transport representation $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, which is a limit diagram if and only if $Q$ is a homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$