We write
\(\mathrm{Cat}_{\infty}^\mathrm{idem}\) for the full subcategory of \(\mathrm{Cat}_{\infty}\) on the idempotent complete small \(\infty\)-categories,
\(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) for the subcategory of \(\mathrm{Cat}_{\infty}^\mathrm{idem}\) on the idempotent complete small \(\infty\)-categories that admit finite coproducts and functors which preserve finite coproducts, and
\(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) for the subcategory of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) on the idempotent complete small \(\infty\)-categories that admit finite colimits and functors which preserve finite colimits.