# Kerodon

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Example 7.5.1.7 (Duality with Homotopy Colimits). Let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $X$ be a Kan complex, and let $X^{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be the diagram of Kan complexes given by the formula $X^{\mathscr {F}}(C) = \operatorname{Fun}( \mathscr {F}(C), X)$. Let us write $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ for the functor given by the formula $\mathscr {F}^{\operatorname{op}}(C) = \mathscr {F}(C)^{\operatorname{op}}$, and let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by $\mathscr {E}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$. Combining Remark 7.5.1.6 with Proposition 5.3.2.21, we obtain canonical isomorphisms

\begin{eqnarray*} \underset {\longleftarrow }{\mathrm{holim}}( X^{\mathscr {F}} )^{\operatorname{op}} & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( \mathscr {E}, X^{\mathscr {F}} )^{\operatorname{op}}_{\bullet } \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, X^{\mathscr {E}} )^{\operatorname{op}}_{\bullet } \\ & \simeq & \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{\operatorname{op}} ), X^{\operatorname{op}} ). \end{eqnarray*}