Proof of Theorem 5.3.5.7.
We first show that $(1)$ implies $(2)$. Assume that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets and let $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ be a scaffold of $U$; we wish to show that $\lambda $ is a categorical equivalence of simplicial sets. By virtue of Corollary 4.5.7.3, it will suffice to show that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map
\[ \Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]
is a categorical equivalence of simplicial sets. We may therefore assume without loss of generality that the category $\operatorname{\mathcal{C}}$ is a linearly ordered set of the form $[n] = \{ 0 < 1 < \cdots < n \} $ for some $n \geq 0$.
We proceed by induction on $n$. If $n = 0$, the result is clear. Let us therefore assume that $n > 0$. Let $S = \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} )$ be the $0$th face of the simplex $\Delta ^ n$ and set $\operatorname{\mathcal{E}}_{+} = S \times _{\Delta ^ n} \operatorname{\mathcal{E}}$. Let $\mathscr {F}_{+}$ denote the restriction of $\mathscr {F}$ to the subcategory $\{ 1 < \cdots < n \} \subset [n]$, so that our inductive hypothesis guarantees that $\lambda $ restricts to a categorical equivalence $\lambda _{+}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+}) \rightarrow \operatorname{\mathcal{E}}_{+}$. Note that Remark 5.3.2.12 supplies an isomorphism of simplicial sets
\[ ( \Delta ^ n \times \mathscr {F}(0) ) {\coprod }_{ (S \times \mathscr {F}(0)) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). \]
Let $V: \Delta ^{n} \rightarrow \Delta ^1$ be the morphism given on vertices by the formula $V(i) = \begin{cases} 0 & \textnormal{ if } i = 0 \\ 1 & \textnormal{ if } i > 0.\end{cases}$ Then $V$ is a cocartesian fibration of simplicial sets, and the edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Delta ^ n$ is $V$-cocartesian. It follows that, for every vertex $x$ of the simplicial set $\mathscr {F}(0)$, the composite map
\[ \Delta ^1 \times \{ x\} \hookrightarrow \Delta ^ n \times \mathscr {F}(0) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow {\lambda } \operatorname{\mathcal{E}} \]
is a $(V \circ U)$-cocartesian edge of $\operatorname{\mathcal{E}}$. Applying Theorem 5.2.4.1 to the cocartesian fibration $V \circ U$, we deduce that the composition
\begin{eqnarray*} ( \Delta ^1 \times \mathscr {F}(0) ) {\coprod }_{ ( \{ 1\} \times \mathscr {F}(0) ) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} ) & \xrightarrow {\iota } & ( \Delta ^{n} \times \mathscr {F}(0) ) {\coprod }_{ (S \times \mathscr {F}(0) ) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_{+} ) \\ & \simeq & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \\ & \xrightarrow {\lambda } & \operatorname{\mathcal{E}}. \end{eqnarray*}
is a categorical equivalence of simplicial sets. Consequently, to show that $\lambda $ is a categorical equivalence of simplicial sets, it will suffice to show that $\iota $ is inner anodyne. By construction, $\iota $ is a pushout of the inclusion map
\[ ( \Delta ^1 {\coprod }_{ \{ 1\} } S ) \times \mathscr {F}(0) \rightarrow \Delta ^ n \times \mathscr {F}(0). \]
By virtue of Lemma 1.5.7.5, it will suffice to show that the inclusion map $\Delta ^1 {\coprod }_{ \{ 1\} } S \hookrightarrow \Delta ^ n$ is inner anodyne. This is a special case of Example 4.3.6.5, since the inclusion $\{ 1\} \hookrightarrow S$ is left anodyne (Lemma 4.3.7.8).
We now show that $(2)$ implies $(1)$. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a functor of $\infty $-categories, and suppose that $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets such that $U \circ \lambda $ is equal to the projection map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. We first claim that $U$ is an isofibration of $\infty $-categories. Since $\operatorname{\mathcal{E}}$ is an $\infty $-category, the morphism $U$ is an inner fibration (Proposition 4.1.1.10). It will therefore suffice to show that, for each object $\widetilde{C} \in \operatorname{\mathcal{E}}$ having image $C = U( \widetilde{C} ) \in \operatorname{\mathcal{C}}$ and every isomorphism $e: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, there exists an isomorphism $\widetilde{e}: \widetilde{C} \rightarrow \widetilde{D}$ in $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{e} ) = e$. Since $\lambda $ is a categorical equivalence, we can choose a vertex $v$ of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and an isomorphism $\widetilde{f}: \widetilde{C} \rightarrow \lambda (v)$ in $\operatorname{\mathcal{E}}$. Let us identify $v$ with a pair $(C',X)$, where $C'$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is a vertex of the simplicial set $\mathscr {F}(C')$. Then $f = U( \widetilde{f} )$ is an isomorphism from $C$ to $C'$ in the category $\operatorname{\mathcal{C}}$. Replacing $v$ by the pair $(C, \mathscr {F}(f^{-1})(X) )$, we can reduce to the case where $C' = C$ and $f = \operatorname{id}_{C}$ so that $\widetilde{f}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. In this case, we can take $\widetilde{e}$ to be any composition of $\widetilde{f}$ with the morphism $\lambda (e, \operatorname{id}_ X ): \lambda (C, X) \rightarrow \lambda (D, \mathscr {F}(e)(X) )$ of $\operatorname{\mathcal{E}}$. This completes the proof that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an isofibration.
Using Corollary 5.3.4.18, we can choose a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ and a scaffold $\lambda ': \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}'$. Then $U'$ is an isofibration, so composition with $\lambda $ induces a categorical equivalence $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) }( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}' )$ (Corollary 4.5.2.34). It follows that there exists a functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ satisfying $U' \circ F = U$ such that $F \circ \lambda $ is isomorphic to $\lambda '$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}' )$. Since $\lambda '$ is a categorical equivalence of simplicial sets, the morphism $F \circ \lambda $ is also a categorical equivalence of simplicial sets (Corollary 4.5.3.9). Applying the two-out-of-three property (Remark 4.5.3.5), we deduce that $F$ is an equivalence of $\infty $-categories. It follows that $U$ is also a cocartesian fibration (Corollary 5.1.6.2) and that $\lambda $ is a scaffold of $U$ (Remark 5.3.4.6).
$\square$