Proposition 7.5.6.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between projectively cofibrant diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\alpha $ is a levelwise categorical equivalence, then the induced map $\varinjlim (\alpha ): \varinjlim ( \mathscr {F} ) \rightarrow \varinjlim ( \mathscr {G} )$ is a categorical equivalence of simplicial sets. If $\alpha $ is a levelwise weak homotopy equivalence, then $\varinjlim (\alpha )$ is a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will prove the first assertion; the second follows by a similar argument. Assume that $\alpha $ is levelwise categorical equivalence and let $\operatorname{\mathcal{D}}$ be an $\infty $-category; we wish to show that precomposition with $\varprojlim (\alpha )$ induces an equivalence of $\infty $-categories $\alpha ^{\ast }: \operatorname{Fun}( \varinjlim ( \mathscr {G} ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \varinjlim ( \mathscr {F}), \operatorname{\mathcal{D}})$. $\alpha $ is a levelwise categorical equivalence, precomposition with $\alpha $ induces a levelwise categorical equivalence $\beta : \operatorname{\mathcal{D}}^{ \mathscr {G} } \rightarrow \operatorname{\mathcal{D}}^{\mathscr {F}}$ in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }})$. Unwinding the definitions, we see that $\alpha ^{\ast }$ can be identified with the limit $\varprojlim (\beta )$. Since $\operatorname{\mathcal{D}}^{\mathscr {F}}$ and $\operatorname{\mathcal{D}}^{\mathscr {G}}$ are isofibrant diagrams (Remark 7.5.6.6), the functor $\varprojlim (\beta )$ is an equivalence of $\infty $-categories (Corollary 4.5.6.17). $\square$