Corollary 5.3.3.25. Let $\operatorname{\mathcal{C}}$ be a category. Then the weighted nerve functor
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]
has a left adjoint, given by the construction $\operatorname{\mathcal{E}}\mapsto \mathscr {G}_{\operatorname{\mathcal{E}}}$ of Notation 5.3.3.23.