Warning 5.6.2.6. In §5.6.1, we introduced a variant of the category of elements construction for contravariant $\mathbf{Cat}$-valued functors $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ (see Definition 5.6.1.4), which is characterized by the formula
\[ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} = ( \int _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F}^{\operatorname{op}} )^{\operatorname{op}}. \]
In the $\infty $-categorical setting, the situation is more subtle: the involution $\operatorname{\mathcal{E}}\mapsto \operatorname{\mathcal{E}}^{\operatorname{op}}$ does not preserve the simplicial structure on the category $\operatorname{QCat}$ and therefore does not induce an involution on the simplicial set $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$. We will return to this point in §
.