Proof.
Fix an integer $n > 0$, and suppose we are given a diagram $\mathscr {F}_0: \operatorname{\mathcal{C}}\star \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{QC}}^{< \kappa }$ for which the restriction $\mathscr {F}_0|_{ \operatorname{\mathcal{C}}\star \{ 0\} }$ coincides with $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }$. We wish to show that $\mathscr {F}_0$ can be extended to a functor $\mathscr {F}: \operatorname{\mathcal{C}}\star \Delta ^ n \rightarrow \operatorname{\mathcal{QC}}^{< \kappa }$. Applying Lemma 5.6.7.1, we can choose a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \ar [r] & \overline{\operatorname{\mathcal{E}}}^{-} \ar [d]^{ \overline{U}^{-} } \\ \operatorname{\mathcal{C}}\star \{ 0\} \ar [r] & \operatorname{\mathcal{C}}\star \operatorname{\partial \Delta }^{n}, } \]
where $\overline{U}^{-}$ is a cocartesian fibration having covariant transport representation $\mathscr {F}_0$. For $0 \leq i \leq n$, let us write $\overline{\operatorname{\mathcal{E}}}^{-}_{i}$ for the $\infty $-category given by the fiber of $\overline{U}^{-}$ on the vertex $i \in \operatorname{\partial \Delta }^{n}$.
Fix an auxiliary symbol $c$, so that the projection map $\operatorname{\mathcal{C}}\rightarrow \{ c\} $ induces a cocartesian fibration of $\infty $-categories $V^{+}: \operatorname{\mathcal{C}}\star \Delta ^{n} \rightarrow \{ c\} \star \Delta ^ n$ (this follows by repeated application of Lemma 5.2.3.17). Note that $V^{+}$ restricts to a morphism of simplicial sets $V^{-}: \operatorname{\mathcal{C}}\star \operatorname{\partial \Delta }^{n} \rightarrow \{ c\} \star \operatorname{\partial \Delta }^{n}$ which is a pullback of $V^{+}$, and therefore also a cocartesian fibration (Remark 5.1.4.6). Applying Proposition 5.1.4.14, we deduce that the composite map $( V^{-} \circ \overline{U}^{-} ): \overline{\operatorname{\mathcal{E}}}^{-} \rightarrow \{ c\} \star \operatorname{\partial \Delta }^{n} \simeq \Lambda ^{n+1}_{0}$ is also a cocartesian fibration.
Let $\mathscr {G}_0: \{ c\} \star \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{QC}}^{< \kappa }$ be a covariant transport representation for the cocartesian fibration $V^{-} \circ \overline{U}^{-}$. Let us identify $\mathscr {G}_{0}(c)$ with the $\infty $-category $\operatorname{\mathcal{E}}$. For $0 \leq i \leq n$, we can identify $\mathscr {G}_{0}(i)$ with the $\infty $-category $\overline{\operatorname{\mathcal{E}}}^{-}_{i}$ for $0 \leq i \leq n$, and the restriction of $\mathscr {G}_{0}$ to the edge $\{ c\} \star \{ i\} $ with a functor $G_ i: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}^{-}_{i}$. Applying Example 7.4.5.9 (and Remark 5.6.5.10), we see that $G_{i}$ is a covariant refraction diagram for the cocartesian fibration
\[ ( \operatorname{\mathcal{C}}\star \{ i\} ) \times _{ \operatorname{\mathcal{C}}\star \operatorname{\partial \Delta }^{n} } \overline{\operatorname{\mathcal{E}}}^{-} \rightarrow \operatorname{\mathcal{C}}\star \{ i\} . \]
In particular, each of the functors $G_{i}$ carries elements of $W$ to isomorphisms in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}^{-}_{i}$ (Remark 7.4.5.10). Moreover, the functor $G_0$ is isomorphic to $\mathrm{Rf}$ (Proposition 7.4.5.8), and therefore exhibits $\overline{\operatorname{\mathcal{E}}}^{-}_{0} = \overline{\operatorname{\mathcal{E}}}_{0}$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$ (Exercise 6.3.1.11). Applying Lemma 7.4.6.1, we can extend $\mathscr {G}_0$ to a diagram $\mathscr {G}: \{ c\} \star \Delta ^{n} \rightarrow \operatorname{\mathcal{QC}}^{<\kappa }$. Using Lemma 5.6.7.1, we can choose a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{\operatorname{\mathcal{E}}}^{-} \ar [r] \ar [d]^{ V^{-} \circ \overline{U}^{-} } & \overline{\operatorname{\mathcal{E}}}^{+} \ar [d]^{ T } \\ \operatorname{\mathcal{C}}\star \operatorname{\partial \Delta }^{n} \ar [r] & \operatorname{\mathcal{C}}\star \Delta ^{n}, } \]
where $T$ is a cocartesian fibration having covariant transport representation $\mathscr {G}$. Note that we can write $T$ uniquely as a composition
\[ \overline{\operatorname{\mathcal{E}}}^{+} \xrightarrow { \overline{U}^{+} } \operatorname{\mathcal{C}}\star \Delta ^{n} \xrightarrow { V^{+} } \{ c\} \star \Delta ^{n}, \]
where $\overline{U}^{+}$ is a morphism of simplicial sets which fits into a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \overline{\operatorname{\mathcal{E}}}^{-} \ar [r] \ar [d]^{ \overline{U}^{-} } & \overline{\operatorname{\mathcal{E}}}^{+} \ar [d]^{ \overline{U}^{+} } \\ \operatorname{\mathcal{C}}\star \operatorname{\partial \Delta }^{n} \ar [r] & \operatorname{\mathcal{C}}\star \Delta ^{n}. } \]
We will show that the morphism $\overline{U}^{+}$ is a cocartesian fibration. Assuming this, we can complete the proof by applying Corollary 5.6.5.13 to extend $\mathscr {F}_0$ to a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\star \Delta ^{n} \rightarrow \operatorname{\mathcal{QC}}^{<\kappa }$ (which is a covariant transport representation for the cocartesian fibration $\overline{U}^{+}$).
We first prove that $\overline{U}^{+}$ is an inner fibration of simplicial sets. Suppose we are given integers $0 < i < m$; we wish to show that every lifting problem
7.53
\begin{equation} \begin{gathered}\label{equation:detection-criterion-lifting} \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{ \sigma _0} \ar [d] & \overline{\operatorname{\mathcal{E}}}^{+} \ar [d]^{ \overline{U}^{+} } \\ \Delta ^{m} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}\star \Delta ^{n} } \end{gathered} \end{equation}
admits a solution. If $\overline{\sigma }$ factors through $\operatorname{\mathcal{C}}$, then a solution exists by virtue of the fact that $U$ is an inner fibration. Let us therefore assume that $\overline{\sigma }$ does not factor through $\operatorname{\mathcal{C}}$. Since $T$ is an inner fibration, we can extend $\sigma _0$ to an $n$-simplex $\sigma $ of $\overline{\operatorname{\mathcal{E}}}^{+}$ satisfying $T \circ \sigma = V^{+} \circ \overline{\sigma }$. We claim that the $n$-simplex $\sigma $ solves the lifting problem (7.53). Set $\overline{\sigma }' = \overline{U}^{+} \circ \sigma $; we wish to show that $\overline{\sigma }'$ coincides with $\overline{\sigma }$ (as $m$-simplices of the simplicial set $\operatorname{\mathcal{C}}\star \Delta ^{n}$). Note that we have $V^{+} \circ \overline{\sigma } = V^{+} \circ \overline{\sigma }'$. It follows that $\overline{\sigma }$ and $\overline{\sigma }'$ both carry the final vertex $m \in \Delta ^{m}$ to the same vertex of $\Delta ^{n} \subseteq \operatorname{\mathcal{C}}\star \Delta ^{n}$. Consequently, it will suffice to show that $\overline{\sigma }$ and $\overline{\sigma }'$ agree when restricted to the face $\Delta ^{m-1} \subseteq \Delta ^{m}$. This follows from the commutativity of the diagram (7.53), since $\Delta ^{m-1}$ is contained in the horn $\Lambda ^{m}_{i} \subseteq \Delta ^{m}$.
Fix an object $X$ of the $\infty $-category $\overline{\operatorname{\mathcal{E}}}^{+}$ having image $\overline{X} = \overline{U}^{+}(X)$ and a morphism $\overline{e}: \overline{X} \rightarrow \overline{Y}$ in the $\infty $-category $\operatorname{\mathcal{C}}\star \Delta ^{m}$. We will complete the proof by showing that $\overline{e}$ can be lifted to an $\overline{U}^{+}$-cocartesian morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$. If $\overline{X}$ and $\overline{Y}$ belong to $\operatorname{\mathcal{C}}$, then we take $e: X \rightarrow Y$ to be a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$ satisfying $U(e) = \overline{e}$ (which exists by virtue of our assumption that $U$ is a cocartesian fibration). Otherwise, we take $e: X \rightarrow Y$ to be a $T$-cocartesian morphism of $\overline{\operatorname{\mathcal{E}}}^{+}$ satisfying $T(e) = V^{+}( \overline{e} )$ (which exists by virtue of the fact that $T$ is a cocartesian fibration). In either case, we will prove that the morphism $e$ is $\overline{U}^{+}$-cocartesian by verifying the criterion of Proposition 5.1.2.1. Choose another object $Z \in \overline{\operatorname{\mathcal{E}}}^{+}$ having image $\overline{Z} = \overline{U}^{+}(Z)$; we wish to show that the diagram of Kan complexes
7.54
\begin{equation} \begin{gathered}\label{equation:diagram-for-checking-cocartesian} \xymatrix@R =50pt@C=50pt{ \{ c\} \times _{ \operatorname{Hom}_{\overline{\operatorname{\mathcal{E}}}^{+}}(X,Y) } \operatorname{Hom}_{\overline{\operatorname{\mathcal{E}}}^{+}}(X,Y,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\overline{\operatorname{\mathcal{E}}}^{+}}(X,Z) \ar [d] \\ \{ \overline{e} \} \times _{ \operatorname{Hom}_{ \operatorname{\mathcal{C}}\star \Delta ^{n} }( \overline{X}, \overline{Y} ) } \operatorname{Hom}_{\operatorname{\mathcal{C}}\star \Delta ^{n}}( \overline{X}, \overline{Y}, \overline{Z} ) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}\star \Delta ^{n}}( \overline{X}, \overline{Z} ) } \end{gathered} \end{equation}
is a homotopy pullback square. We consider several cases:
Suppose first that the object $\overline{Z}$ belongs to $\operatorname{\mathcal{C}}$. If $\overline{X}$ and $\overline{Y}$ belong to $\operatorname{\mathcal{C}}$, then we deduce that (7.54) is a homotopy pullback square by applying Proposition 5.1.2.1 to the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (since, by construction, the morphism $e$ is $U$-cocartesian). Otherwise, each of the Kan complexes appearing in the diagram (7.54) is empty, so there is nothing to prove.
Suppose that the objects $\overline{Y}$ and $\overline{Z}$ belong to $\Delta ^{n}$. In this case, we deduce that (7.54) is a homotopy pullback square by applying Proposition 5.1.2.1 to the cocartesian fibration $T: \overline{\operatorname{\mathcal{E}}}^{+} \rightarrow \{ c\} \star \Delta ^ n$ (since, by construction, the morphism $e$ is $T$-cocartesian).
Suppose that the objects $\overline{X}$ and $\overline{Y}$ belong to $\operatorname{\mathcal{C}}$, but the object $\overline{Z}$ belongs to $\Delta ^{n}$. In this case, the Kan complexes on the bottom row of (7.54) are contractible (see Example 4.6.1.6; in fact, they are both isomorphic to $\Delta ^{0}$). In particular, the bottom horizontal map is a homotopy equivalence. To show that (7.54) is a homotopy pullback square, we must show that the upper horizontal map is also a homotopy equivalence (Corollary 3.4.1.5). In other words, we must show that composition with the homotopy class $[e]$ induces an isomorphism $\theta : \operatorname{Hom}_{\overline{\operatorname{\mathcal{E}}}^{+} }( Y, Z) \rightarrow \operatorname{Hom}_{ \overline{\operatorname{\mathcal{E}}}^{+} }(X, Z )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Notation 4.6.9.15). Let
\[ G: \operatorname{\mathcal{E}}= \{ c\} \times _{ \{ c\} \star \Delta ^{n} } \overline{\operatorname{\mathcal{E}}}^{+} \rightarrow \{ \overline{Z} \} \times _{ \{ c\} \star \Delta ^{n} } \overline{\operatorname{\mathcal{E}}}^{+} = \overline{\operatorname{\mathcal{E}}}^{+}_{ \overline{Z} } \]
be given by covariant transport for the cocartesian fibration $T$. Using Corollary 5.1.2.3, we can identify $\theta $ with the morphism $\operatorname{Hom}_{ \overline{\operatorname{\mathcal{E}}}^{+}_{ \overline{Z} }}( G(Y), Z) \rightarrow \operatorname{Hom}_{ \overline{\operatorname{\mathcal{E}}}^{+}_{ \overline{Z} }}( G(X), Z)$ given by precomposition with the morphism $G(e): G(X) \rightarrow G(Y)$. Since the morphism $e$ is $U$-cocartesian, its image $G(e)$ is an isomorphism in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}^{+}_{\overline{Z}}$, so that $\theta $ is a homotopy equivalence as desired.
$\square$