$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 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.

Proof. By virtue of Proposition, 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 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 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, and therefore essentially $\kappa $-small. $\square$