Kerodon

Corollary 7.4.3.4. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a (strictly commutative) diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. Then a Kan complex $X$ is a colimit of the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{S}}$ if and only if it is weakly homotopy equivalent to the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Definition 5.3.3.1.