Remark 8.6.8.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration which is conjugate to $U$. By definition, a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a contravariant transport representation for $U^{\dagger }$ if and only if it is a covariant transport representation for $U$. In this case:
Proposition 7.4.4.1 asserts that the limit $\varprojlim (\mathscr {F})$ can be computed as the $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$.
Proposition 8.6.8.11 asserts that the limit $\varprojlim (\mathscr {F})$ can be computed as the $\infty $-category $\operatorname{Fun}^{\operatorname{Cart}}_{/ \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\dagger })$ of cartesian sections of $\operatorname{\mathcal{E}}^{\dagger }$.
To identify these conditions, it suffices to observe that the $\infty $-categories $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}^{\operatorname{Cart}}_{/ \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\dagger })$ are equivalent. In fact, we can be more precise. Assume for simplicity that $\operatorname{\mathcal{C}}$ is an $\infty $-category, and fix a functor $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$ (Definition 8.6.1.1). Then $T$ determines a canonical equivalence $\Theta : \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}^{\operatorname{Cart}}_{/ \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\dagger })$, which is characterized (up to isomorphism) by the requirement that the diagram of $\infty $-categories
commutes up to isomorphism. Here the left vertical map is given by precomposition with the projection $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$, and the right vertical map is given by pullback along the projection $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ followed by postcomposition with $T$. See Corollary 8.6.2.12 for a more general assertion.