Kerodon

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Corollary 7.5.3.5 (Existence of Isofibrant Replacements). 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 categorical equivalence and $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram of $\infty $-categories.

Proof. Using Proposition 4.1.3.2, we can reduce to the case where $\mathscr {F}$ is a diagram of $\infty $-categories. In this case, we can take $\alpha $ to be the natural transformation $\mathscr {F} \hookrightarrow \mathscr {F}^{+}$ of Construction 7.5.3.3 (Proposition 7.5.3.4). $\square$