Corollary 5.3.2.24. Let $\operatorname{\mathcal{C}}$ be a small category. Then the homotopy colimit functor
\[ \underset { \longrightarrow }{\mathrm{holim}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]
admits a right adjoint, given by the functor
\[ (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \mapsto (\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}) \]
of Construction 5.3.1.1.