Corollary 7.5.5.15. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be a functor. Let $\overline{ \mathscr {F} }^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ 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 categorical limit diagram if and only if $\overline{\mathscr {F}}^{\operatorname{op}}$ is a categorical limit diagram.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Proposition 4.1.3.2, we can choose a functor $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$. By virtue of Corollary 7.5.5.14, it will suffice to show that $\overline{\mathscr {G}}$ is a categorical limit diagram if and only if $\overline{\mathscr {G}}^{\operatorname{op}}$ is a categorical limit diagram. This follows by combining Proposition 7.5.5.7 with Corollary 7.5.4.12. $\square$