Corollary 5.3.3.20. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {F}'$ be a natural transformation between functors $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then $\alpha $ is a levelwise categorical equivalence if and only if the induced map $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ is a categorical equivalence of simplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume first that $\alpha $ is a levelwise categorical equivalence. To prove that $T$ is a categorical equivalence of simplicial sets, it will suffice to show that for every simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map $T_{\sigma }: \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ is a categorical equivalence of simplicial sets (Corollary 4.5.7.3). Using Remark 5.3.3.7, we can reduce to the special case where $\operatorname{\mathcal{C}}$ is the linearly ordered $[n] = \{ 0 < 1 < \cdots < n \} $ for some $n \geq 0$. We now proceed by induction on $n$. If $n = 0$, the result is immediate from Example 5.3.3.2. The inductive step follows by combining Example 5.3.3.12 with Corollary 5.2.4.7.
We now prove the converse. Using Proposition 4.1.3.2, we can choose a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F} \ar [r]^{\alpha } \ar [d] & \mathscr {F}' \ar [d] \\ \mathscr {G} \ar [r]^{\beta } & \mathscr {G}' } \]
in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, where the vertical maps are levelwise categorical equivalences and the simplicial sets $\mathscr {G}(C)$ and $\mathscr {G}'(C)$ are $\infty $-categories for each $C \in \operatorname{\mathcal{C}}$. Using the first part of the proof, we can replace $\alpha $ by $\beta $ and thereby reduce to the special case where $\mathscr {F}$ and $\mathscr {F}'$ are diagrams of $\infty $-categories. In this case, the projection maps $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \leftarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ are cocartesian fibrations of $\infty $-categories (Corollary 5.3.3.16). It then follows from Theorem 5.1.6.1 (together with Example 5.3.3.8) that if $T$ is an equivalence of $\infty $-categories, then $\alpha $ is a levelwise categorical equivalence. $\square$