Variant 7.5.3.6. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a monomorphism of diagrams $\alpha : \mathscr {F} \hookrightarrow \mathscr {G}$, where $\alpha $ is a levelwise weak homotopy equivalence and $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ is an isofibrant diagram of Kan complexes.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Proposition 3.1.7.1, we can reduce to the case where $\mathscr {F}$ is a diagram of Kan complexes. In this case, we can again take $\alpha $ to be the natural transformation $\mathscr {F} \hookrightarrow \mathscr {F}^{+}$ of Construction 7.5.3.3 (since $\alpha $ is a levelwise categorical equivalence, it follows that $\mathscr {F}^{+}$ is also a diagram of Kan complexes: see Remark 4.5.1.21). $\square$