Proof of Proposition 4.5.5.1.
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume first that $F$ satisfies condition $(\ast )$ of Proposition 4.5.5.1; we will prove that $F$ is an isofibration. For $0 < i < n$, the inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a categorical equivalence (Corollary 4.5.3.14), so condition $(\ast )$ guarantees that $F$ is an inner fibration. Fix an object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $u: D \rightarrow F(C)$ in the $\infty $-category $\operatorname{\mathcal{D}}$; we wish to show that $u$ can be lifted to an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$. By virtue of Corollary 4.4.3.14, we can assume that $u = G(v)$ for some functor $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{E}}$ is a contractible Kan complex and $v: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{E}}$. Since the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence (Example 4.5.1.13), condition $(\ast )$ guarantees the existence of a solution to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ Y\} \ar [r]^-{ Y \mapsto C} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{E}}\ar [r]^-{G} \ar@ {-->}[ur]^{\overline{G}} & \operatorname{\mathcal{D}}. } \]
Then $\overline{u} = \overline{G}(v)$ is an isomorphism of $\operatorname{\mathcal{C}}$ having the desired property.
Now suppose that the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isofibration; we wish to show that condition $(\ast )$ is satisfied. Let $B$ be a simplicial set and $A \subseteq B$ a simplicial subset for which the inclusion $A \hookrightarrow B$ is a categorical equivalence. We wish to show that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r]^-{f_0} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar [r]^-{\overline{f}} \ar@ {-->}[ur]^{f} & \operatorname{\mathcal{D}}} \]
admits a solution. Invoking Lemma 4.5.5.2, we see that $f_0$ can be extended to a morphism of simplicial sets $f': B \rightarrow \operatorname{\mathcal{C}}$. Let $\overline{f}'$ denote the composition $B \xrightarrow {f'} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$, so that $\overline{f}|_{A} = \overline{f}'|_{A}$. Invoking Lemma 4.5.5.3, we conclude that there exists an isomorphism $\overline{u}: \overline{f} \rightarrow \overline{f}'$ in the diagram $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}(A, \operatorname{\mathcal{D}})$ is the identity transformation $\operatorname{id}_{ \overline{f}|_{A} }$. Applying Corollary 4.4.5.9, we deduce that $\overline{u}$ can be lifted to an isomorphism $u: f \rightarrow f'$ in the diagram $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ whose image in $\operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is the identity transformation $\operatorname{id}_{ f_0 }$. The diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ then satisfies $f|_{A} = f_0$ and $F \circ f = \overline{f}$, as desired.
$\square$