Kerodon

Proof. Assume first that the inner fibration $U \circ V$ is essentially small, and choose an $n$-simplex $\widetilde{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$; we wish to show that the fiber product $\Delta ^ n \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. To prove this, we are free to replace $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ by $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, respectively, and thereby reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$. In this case, the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are essentially $\kappa $-small, so the desired result follows from Example 4.7.9.5.

We now prove the converse. Assume that $\kappa $ is regular, that $V$ is an essentially $\kappa $-small isofibration, and choose an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. This follows by applying Corollary 4.7.9.12 to the isofibration of $\infty $-categories $(\operatorname{id}\times V): \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, which is essentially $\kappa $-small by virtue of Remark 4.7.9.3. $\square$