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

Corollary Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and suppose that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. The following conditions are equivalent:


The $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small.


For every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small.

Proof. Using Corollary, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is an $\infty $-category. Using Proposition, we can write $U$ as the pullback of a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$. Proposition then guarantees that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is a categorical equivalence of simplicial sets. Since the inclusion $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ is bijective on vertices, every fiber of $U'$ can also be regarded as a fiber of $U$. We can therefore replace $U$ by $U'$, and thereby reduce to proving Corollary in the special case where $U$ is a cocartesian fibration of $\infty $-categories.

Note that $U$ is an isofibration (Proposition Consequently, for each object $C \in \operatorname{\mathcal{C}}$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \{ C\} \ar [r] & \operatorname{\mathcal{C}}} \]

is a categorical pullback diagram of simplicial sets (Corollary The implication $(1) \Rightarrow (2)$ now follows from Corollary (and does not require the assumption that $\kappa $ is regular). To prove the reverse implication, we first note that the $\infty $-category $\operatorname{\mathcal{C}}$ and each fiber $\operatorname{\mathcal{E}}_{C}$ are locally $\kappa $-small (Example Applying Proposition, we see that $\operatorname{\mathcal{E}}$ is locally $\kappa $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}^{\simeq } )$ is $\kappa $-small (Proposition The functor $U$ induces a map $\theta : \pi _0( \operatorname{\mathcal{E}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$, whose target is $\kappa $-small. Invoking the regularity of $\kappa $, we are reduced to showing that for every element $[C] \in \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$, the inverse image $\theta ^{-1} \{ [C] \} $ is a $\kappa $-small set. Let us identify $[C]$ with the isomorphism class of an object $C \in \operatorname{\mathcal{C}}$. Then there is a surjective map $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } ) \twoheadrightarrow \theta ^{-1} \{ [C] \} $. Since the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is essentially $\kappa $-small, the set $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } )$ is $\kappa $-small, so that the quotient $\theta ^{-1} \{ [C] \} $ is also $\kappa $-small (Remark $\square$