Kerodon

Proof. Assume that $(2)$ is satisfied; we will prove $(1)$ (the reverse implication is immediate from the definitions). Fix an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small: that is, that the morphism space $M = \operatorname{Hom}_{\operatorname{\mathcal{E}}_{\sigma }}(X,Y)$ is essentially $\kappa $-small for every pair of objects $X,Y \in \operatorname{\mathcal{E}}_{\sigma }$. Let $i,j \in [n]$ be the images of $X$ and $Y$ under the projection map $\operatorname{\mathcal{E}}_{\sigma } \rightarrow \Delta ^ n$. We may assume without loss of generality that $i \leq j$ (otherwise, the morphism space $M$ is empty). In this case, $M$ can also be identified with a morphism space in the $\infty $-category $\operatorname{\mathcal{E}}_{e}$, where $e$ is the edge of $\operatorname{\mathcal{C}}$ given by the composite map $\Delta ^{1} \xrightarrow { (i,j) } \Delta ^ n \xrightarrow {\sigma } \operatorname{\mathcal{C}}$, so the desired result follows from assumption $(2)$. $\square$