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Example 7.5.2.9 (Duality with Homotopy Colimits). Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $W$ denote the collection of horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{\operatorname{op}} )$ (see Definition 5.3.4.1). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}^{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given by the formula $\operatorname{\mathcal{D}}^{\mathscr {F}}(C) = \operatorname{Fun}( \mathscr {F}(C), \operatorname{\mathcal{D}})$. Arguing as in Example 7.5.1.7, we obtain a canonical isomorphism

\[ \theta : \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\operatorname{\mathcal{D}}^{\mathscr {F}} }(\operatorname{\mathcal{C}}) )^{\operatorname{op}} \simeq \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}^{\operatorname{op}}), \operatorname{\mathcal{D}}^{\operatorname{op}} ). \]

Unwinding the definitions, we see that $\theta $ restricts to an isomorphism of $\infty $-categories $ \underset {\longleftarrow }{\mathrm{holim}}( \operatorname{\mathcal{D}}^{\mathscr {F}} )^{\operatorname{op}} \simeq \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}^{\operatorname{op}})[W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}} )$.