Remark 5.3.0.1. There is a close analogy between the homotopy colimit construction (studied in §5.3.2) and the weighted nerve construction (studied in §5.3.3).
The formation of homotopy colimits determines a functor
\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ). \]This functor has a right adjoint, which carries an object $\operatorname{\mathcal{E}}\in (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the diagram
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}). \]See Corollary 5.3.2.24.
The formation of weighted nerves determines a functor
\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}). \]This functor has a left adjoint, which carries an object $\operatorname{\mathcal{E}}\in (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the diagram
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{/C}) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}. \]See Corollary 5.3.3.25.