Corollary 5.3.2.22. Let $\operatorname{\mathcal{C}}$ be a small category. Then the homotopy colimit functor
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \]
admits a right adjoint, given by the construction
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \quad \quad \operatorname{\mathcal{E}}\mapsto (C \mapsto \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}) ). \]