Kerodon

Remark 8.6.5.7. Construction 8.6.5.6 is independent of the choice of the cardinal $\kappa $, provided that each of the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ is locally $\kappa $-small. If this condition is satisfied and $\lambda \geq \kappa $, then every corepresentable functor $\mathscr {F}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ factors through $\operatorname{\mathcal{S}}^{< \kappa }$. It follows that $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) = \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$.