Remark 4.5.5.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We have now given two a priori different definitions of an isofibration from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$:
According to Definition 4.4.1.4, an isofibration $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration with the property that every isomorphism $u: D \rightarrow F(C)$ in the $\infty $-category $\operatorname{\mathcal{D}}$ can be lifted to an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
According to Definition 4.5.5.5, an isofibration $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism of simplicial sets which has the right lifting property with respect to all monomorphisms $A \hookrightarrow B$ which are categorical equivalences.
However, these definitions are equivalent: this is the content of Proposition 4.4.5.1.