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Warning 4.7.9.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. If $U$ is essentially $\kappa $-small, then the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small for each object $C \in \operatorname{\mathcal{C}}$. Beware that the converse is false in general. For example, let $S$ be a set and let $\operatorname{\mathcal{E}}$ be the category containing a pair of objects $X$ and $Y$, with morphisms given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( X, X) = \{ \operatorname{id}_{X} \} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( Y, Y ) = \{ \operatorname{id}_{Y} \} \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) = S \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,X) = \emptyset . \]

Then there is a unique isofibration $U: \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \Delta ^1$ satisfying $U(X) = 0$ and $U(Y) = 1$. The fibers $U^{-1} \{ 0\} $ and $U^{-1} \{ 1\} $ are isomorphic to $\Delta ^0$ (and are therefore essentially $\kappa $-small for every uncountable cardinal $\kappa $). However, the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ is essentially $\kappa $-small if and only if the set $S$ is $\kappa $-small.