Kerodon

Corollary 7.5.7.8. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a functor. Let $\overline{ \mathscr {F} }^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be the functor given on objects by $\overline{ \mathscr {F} }^{\operatorname{op}}(C) = \overline{\mathscr {F}}(C)^{\operatorname{op}}$. Then $\overline{\mathscr {F}}$ is a homotopy colimit diagram if and only if $\overline{\mathscr {F}}^{\operatorname{op}}$ is a homotopy colimit diagram.