Exercise 7.5.2.15 (Homotopy Limits of Sets). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram in the category of sets. Let us abuse notation by identifying $\operatorname{Set}$ with the full subcategory of $\operatorname{Kan}$ spanned by the constant simplicial sets. Show that the comparison map $\varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Remark 7.5.2.12 is an isomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$