Corollary 6.2.5.5 (06MB)

Corollary 6.2.5.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. The following conditions are equivalent:

$(1)$: The morphism $U$ is a locally cocartesian fibration and, for every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ admits a right adjoint.
$(2)$: The morphism $U$ is a locally cartesian fibration and, for every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ admits a left adjoint.

Moreover, if these conditions are satisfied, then the functor $e_{!}$ is left adjoint to $e^{\ast }$ for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$. In this case, the implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.5.4. The converse follows by applying the same argument to the inner fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$