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Proposition 8.6.1.8 (Conjugates of Left Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of simplicial sets. Then:

$(a)$

The map $H: \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ is a trivial Kan fibration of simplicial sets.

$(b)$

Let $T_0$ be a section of $H$, and let $T: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be the composition of $T_0$ with the projection map $\operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$. Then $T$ exhibits the opposite fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of $U$.

Proof. Note that the morphism $H$ factors as a composition

$\operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}),$

where the map on the left is a left fibration by virtue of Proposition 8.1.1.15, and the map on the right is pullback of $U$ (and is therefore also a left fibration). It follows that $H$ is a left fibration (Remark 4.2.1.11). To prove $(a)$, it will suffice to show that every fiber of $H$ is a contractible Kan complex (Proposition 4.4.2.14). For this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^0$. In this case, $\operatorname{\mathcal{E}}$ is a Kan complex (Proposition 4.4.2.1) and we can identify $H$ with the projection map $\operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$, which is a trivial Kan fibration by virtue of Corollary 8.1.2.3.

Let $T: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be as in $(b)$; we wish to show that $T$ satisfies conditions $(0)$, $(1)$, and $(2)$ of Definition 8.6.1.1. Condition $(0)$ follows from the commutativity of the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{E}}) \ar [rr] \ar [d]^{H} & & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}, }$

and condition $(2)$ is vacuous (our assumption that $U$ is a left fibration guarantees that every edge of $\operatorname{\mathcal{E}}$ is $U$-cocartesian; see Example 5.1.1.3). To verify condition $(1)$, we may again assume that $\operatorname{\mathcal{C}}= \Delta ^0$, in which case the desired result follows from the observation that the projection maps $\operatorname{\mathcal{E}}^{\operatorname{op}} \leftarrow \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ are homotopy equivalences of Kan complexes (see Corollary 8.1.2.3). $\square$