Proposition 4.7.5.15. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\kappa $ be an uncountable cardinal. If $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, then the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is also essentially $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose equivalences of $\infty $-categories
\[ \operatorname{\mathcal{D}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \quad \quad \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\quad \quad \operatorname{\mathcal{D}}_1 \rightarrow \operatorname{\mathcal{C}}_1, \]
where $\operatorname{\mathcal{D}}_0$, $\operatorname{\mathcal{D}}_1$, and $\operatorname{\mathcal{D}}$ are $\kappa $-small. By virtue of Remark 4.6.4.4, the induced maps
\[ \operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \leftarrow \operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_1 \rightarrow \operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1 \]
are equivalences of $\infty $-categories. It will therefore suffice to show that the $\infty $-category $\operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ is $\kappa $-small. This follows from Corollaries 4.7.4.14 and 4.7.4.12, since $\operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ can be identified with a simplicial subset of the product $\operatorname{\mathcal{D}}_0 \times \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{D}}_1$. $\square$