Kerodon

Remark 9.5.5.4. In the situation of Proposition 9.5.5.3, the $\infty $-category $\operatorname{\mathcal{C}}_0$ is unique up to Morita equivalence (see Proposition 9.4.1.19). If $\kappa $ is uncountable, the assumption that $\operatorname{\mathcal{C}}_0$ is $\kappa $-cocomplete guarantees that it is idempotent-complete (Corollary 8.5.4.19), so $\operatorname{\mathcal{C}}_0$ is unique up to equivalence. In the case $\kappa = \aleph _0$, this is not necessarily true. For example, if $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ is the $\infty $-category of spaces, then we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{S}}_{\mathrm{fin}}$ to be the full subcategory spanned by the essentially finite spaces (Proposition 9.3.2.16), which is finitely cocomplete but not idempotent-complete (Proposition 9.2.6.3 and Warning 9.2.6.8).