Variant 6.3.2.6. Let $\kappa $ be an uncountable cardinal, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of edges $W$ (Definition 6.3.1.9). If $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, then $\operatorname{\mathcal{D}}$ is essentially $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Without loss of generality, we may assume that $F$ is a monomorphism of simplicial sets. Choose a categorical equivalence of simplicial sets $u: \operatorname{\mathcal{C}}\rightarrow \overline{\operatorname{\mathcal{C}}}$, where $\overline{\operatorname{\mathcal{C}}}$ is $\kappa $-small, and form a pushout diagram of simplicial sets
6.11
\begin{equation} \begin{gathered}\label{equation:localization-essentially-small} \xymatrix { \operatorname{\mathcal{C}}\ar [r]^{u} \ar [d]^{F} & \overline{\operatorname{\mathcal{C}}} \ar [d]^{\overline{F}} \\ \operatorname{\mathcal{D}}\ar [r]^{v} & \overline{\operatorname{\mathcal{D}}} } \end{gathered} \end{equation}
Then (6.11) is a categorical pushout square (Example 4.5.4.12), so $v$ is also a categorical equivalence (Proposition 4.5.4.10). Moreover, the morphism $\overline{F}$ exhibits $\overline{D}$ as a localization of $\overline{\operatorname{\mathcal{C}}}$ with respect to $u(W)$ (Corollary 6.3.4.3). We may therefore replace $F$ by $\overline{F}$, and thereby reduce to proving Variant 6.3.2.6 in the special case where $\operatorname{\mathcal{C}}$ is $\kappa $-small. In particular, the set of edges $W$ is $\kappa $-small.
Let $Q$ be a contractible Kan complex which is equipped with a monomorphism $\Delta ^1 \hookrightarrow Q$ and has only countably many simplices. Form a pushout diagram of simplicial sets
\[ \xymatrix { \coprod _{w \in W} \Delta ^1 \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{G} \\ \coprod _{w \in W} Q \ar [r] & \operatorname{\mathcal{C}}', } \]
so that $\operatorname{\mathcal{C}}'$ is $\kappa $-small (Remark 4.7.4.6). It follows from Corollary 6.3.4.3 that the morphism $G$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Using Proposition 4.7.5.5, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}''$, where $\operatorname{\mathcal{C}}''$ is a $\kappa $-small $\infty $-category. Then $\operatorname{\mathcal{C}}''$ is also a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, so Remark 6.3.2.2 supplies a categorical equivalence of simplicial sets $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}''$. It follows that $\operatorname{\mathcal{D}}$ is essentially $\kappa $-small, as desired. $\square$