Remark 7.5.2.12 (03AH): Comparison with the Limit

Remark 7.5.2.12 (Comparison with the Limit). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories and let $X = \varprojlim ( \mathscr {F} )$ denote the limit of $\mathscr {F}$, formed in the category of simplicial sets. Let $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the constant functor taking the value $X$. We then have a tautological map $\underline{X} \rightarrow \mathscr {F}$. The induced morphism of simplicial sets

\[ X \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{ \underline{X} }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]

determines a comparison map $\iota : X = \varprojlim ( \mathscr {F} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$. Note that $\iota $ is a monomorphism of simplicial sets (since each of the projection maps $X = \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$ factor through $\iota $).