Remark 9.5.4.16 (Ambidexterity). Suppose we are given a small diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$, which we identify with the covariant transport representation of a presentable fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Using Corollary 7.4.4.2 and Remark 9.5.4.15, we see that the limit and colimit of $\mathscr {F}$ are given by the $\infty $-categories $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian and cartesian sections of $U$, respectively. If $\operatorname{\mathcal{C}}$ is a Kan complex, every section of $U$ is both cartesian and cocartesian. In this case, we obtain a canonical equivalence
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
\[ \varprojlim ( \mathscr {F} ) \simeq \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) = \operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) = \operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \simeq \varinjlim (\mathscr {F} ). \]