Warning 9.2.9.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be as in Proposition 9.2.9.14 and let $X$ be an object of $\operatorname{\mathcal{E}}$. If $U(X) = 1$, then $X$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{E}}$ if and only if it is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{E}}_{1}$. If $U(X) = 0$, the analogous statement is not necessarily true. In this case, Remark 9.2.9.13 shows that $X$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}$ if and only if it is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_0$ and satisfies the following additional condition:
- $(\star )$
The functor
\[ \operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{E}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X, \bullet ) } \operatorname{\mathcal{S}}_{< \mu } \]is $(\kappa ,\lambda )$-finitary. Here $\mu \geq \lambda $ is any infinite cardinal of cofinality $\geq \lambda $ for which $\operatorname{\mathcal{E}}$ is locally $\mu $-small (so that the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X, \bullet ): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is well-defined).
Note that condition $(\star )$ is automatic in either of the following situations:
The functor $U$ is a cartesian fibration and the contravariant transport functor $\operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$ is $(\kappa ,\lambda )$-finitary.
There exists a $U$-cocartesian morphism $f: X \rightarrow Y$, where $Y$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{E}}_1$.