Kerodon

Remark 4.5.2.24. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories. Applying Corollary 4.5.2.23, we can factor $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}\xrightarrow {F''} \operatorname{\mathcal{E}}$, where $F''$ is an isofibration and $F'$ is an equivalence of $\infty $-categories. For each object $E \in \operatorname{\mathcal{E}}$, the equivalence $F'$ restricts to a functor $F'_{E}: \operatorname{\mathcal{C}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E}$. Beware that $F'_{E}$ need not be an equivalence of $\infty $-categories. However, it is always fully faithful: see Proposition 4.6.2.8.