Kerodon

Remark 8.6.9.10. In the statement of Theorem 8.6.9.9, we have implicitly assumed that the $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{E}}$, and $\operatorname{\mathcal{E}}'$ are essentially small, so that the transport representations $\mathscr {F}$, $\mathscr {F}'$ and the coend $\operatorname{Coend}( \mathscr {F}, \mathscr {F}' ) \in \operatorname{\mathcal{QC}}$ are well-defined. More generally, if $\kappa $ is an uncountable regular cardinal such that $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{E}}$, and $\operatorname{\mathcal{E}}'$ are essentially $\kappa $-small, then we can regard $( \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}' )[W^{-1} ]$ as a coend of $\mathscr {F}$ against $\mathscr {F}'$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \kappa }$.