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Corollary Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a projectively cofibrant diagram of simplicial sets. Then the comparison map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ of Remark is a weak homotopy equivalence.

Proof. By virtue of Remark, it will suffice to show that the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ of Construction induces a weak homotopy equivalence $\varinjlim (\alpha ): \varinjlim ( \mathscr {F}_{+} ) \rightarrow \varinjlim ( \mathscr {F} )$. This is a special case of Proposition, since $\alpha $ is a levelwise weak homotopy equivalence between projectively cofibrant diagrams (Proposition $\square$