Proposition 7.5.3.7. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram, and let $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the isofibrant replacement of Construction 7.5.3.3. Then there is a canonical isomorphism of simplicial sets $\theta : \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow {\sim } \varprojlim (\mathscr {F}^{+} )$, which is characterized by the following requirement: for each object $C \in \operatorname{\mathcal{C}}$, the composition
\begin{eqnarray*} \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) ) & = & \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \\ & \xrightarrow {\theta } & \varprojlim ( \mathscr {F}^{+} ) \\ & \rightarrow & \mathscr {F}^{+}(C) \\ & = & \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{C/}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) ) \end{eqnarray*}
is given by precomposition with the projection map $\operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{C}}$.