Proposition 8.5.1.15. Let $\operatorname{\mathcal{X}}$ and $\operatorname{\mathcal{Y}}$ be $\infty $-categories and let $\kappa $ be an uncountable cardinal. Suppose that $\operatorname{\mathcal{Y}}$ is a retract of $\operatorname{\mathcal{X}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$. If $\operatorname{\mathcal{X}}$ is essentially $\kappa $-small, then $\operatorname{\mathcal{Y}}$ is also essentially $\kappa $-small.
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Proof. By virtue of Proposition 4.7.6.15, we may assume that the $\infty $-categories $\operatorname{\mathcal{X}}$ and $\operatorname{\mathcal{Y}}$ are minimal, so that $\operatorname{\mathcal{X}}$ is a $\kappa $-small simplicial set (Corollary 4.7.6.12). Choose functors $i: \operatorname{\mathcal{Y}}\rightarrow \operatorname{\mathcal{X}}$ and $r: \operatorname{\mathcal{X}}\rightarrow \operatorname{\mathcal{Y}}$ such that the composition $(r \circ i): \operatorname{\mathcal{Y}}\rightarrow \operatorname{\mathcal{Y}}$ is isomorphic to the identity functor. Then $r \circ i$ is an equivalence of $\infty $-categories. Since $\operatorname{\mathcal{Y}}$ is minimal, it follows that $r \circ i$ is an isomorphism of simplicial sets (Proposition 4.7.6.13). In particular, the functor $i: \operatorname{\mathcal{Y}}\rightarrow \operatorname{\mathcal{X}}$ is a monomorphism of simplicial sets. It follows that $\operatorname{\mathcal{Y}}$ is $\kappa $-small (Remark 4.7.4.8), and therefore essentially $\kappa $-small. $\square$