Proposition 6.2.3.4. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category equipped with a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Let $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ be a functor given by covariant transport along the nondegenerate edge $e$ of $\Delta ^1$. Then the functor $F$ admits a right adjoint if and only if $U$ is a cartesian fibration. In this case, the right adjoint to $F$ is given by contravariant transport along $e$.
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Proof. Let $\iota _0: \operatorname{\mathcal{E}}_0 \hookrightarrow \operatorname{\mathcal{E}}$ and $\iota _1: \operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{E}}$ denote the inclusion maps. Since $U$ is a cocartesian fibration, $\operatorname{\mathcal{E}}_1$ is a reflective subcategory of $\operatorname{\mathcal{E}}$ (Corollary 6.2.3.2). Let $L: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_1$ be a $\operatorname{\mathcal{E}}_1$-reflection functor (Lemma 6.2.2.16). Without loss of generality, we may assume that the functor $F: \operatorname{\mathcal{E}}_{0} \rightarrow \operatorname{\mathcal{E}}_1$ factors as a composition $\operatorname{\mathcal{E}}_0 \xrightarrow {\iota _0} \operatorname{\mathcal{E}}\xrightarrow {L} \operatorname{\mathcal{E}}_1$ (Remark 6.2.3.3). Note that $L$ is a left adjoint to the inclusion $\iota _1: \operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{E}}$ (Proposition 6.2.2.17).
Suppose that $U$ is also a cartesian fibration, so that the subcategory $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ is coreflective (Corollary 6.2.3.2). Let $L': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_0$ be a $\operatorname{\mathcal{E}}_0$-coreflection functor (Corollary 6.2.3.2), so that $L'$ can be regarded as a right adjoint to $\iota _0$ (Proposition 6.2.2.17). Invoking Remark 6.2.1.8, we conclude that the composite functor $F = L \circ \iota _0$ has a right adjoint $G$, given by the composition $L' \circ \iota _1 = L'|_{\operatorname{\mathcal{E}}_1}$. Moreover, Remark 6.2.3.3 guarantees that $G: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$ is given by contravariant transport along $e$.
We now prove the converse. Suppose that the functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ admits a right adjoint $G: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$. Fix an object $Z \in \operatorname{\mathcal{E}}_1$; we wish to show that there exists an object $Y \in \operatorname{\mathcal{E}}_0$ and a $U$-cartesian morphism $f: Y \rightarrow Z$. Let $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{E}}_1}$ be the counit of an adjunction between $F$ and $G$. Set $Y = G(Z)$, so that $\epsilon $ determines a morphism $\epsilon _{Z}: F(Y) \rightarrow Z$ in the $\infty $-category $\operatorname{\mathcal{E}}_1$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow L$ be a natural transformation which exhibits $L$ as a $\operatorname{\mathcal{E}}_1$-reflection functor, so that $\eta $ determines a morphism $\eta _{Y}: Y \rightarrow F(Y)$. Let $f: Y \rightarrow Z$ be a composition of $\eta _{Y}$ with $\epsilon _{Z}$. We will complete the proof by showing that $f$ is $U$-cartesian. To prove this, it will suffice to show that for every object $X \in \operatorname{\mathcal{E}}_0$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}_0}( X, Y) \xrightarrow { [\eta _ Y] \circ } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X, F(Y) ) = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( X, (F \circ G)(Z)) \xrightarrow { [\epsilon _ Z] \circ } \operatorname{Map}_{\operatorname{\mathcal{E}}}( X, Z) \]
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Corollary 5.1.2.3). Unwinding the definitions, we see that this map factors as a composition
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}_0}( X, G(Z) ) \xrightarrow {F} \operatorname{Hom}_{\operatorname{\mathcal{E}}_1}( F(X), (F \circ G)(Z) ) \xrightarrow { [ \epsilon _ Z ] \circ } \operatorname{Hom}_{\operatorname{\mathcal{E}}_1}( F(X), Z ) \xrightarrow { \circ [\eta _ X] } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z), \]
where the composition of the first two maps is an isomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$ because $\epsilon $ is the counit of an adjunction (see Proposition 6.2.1.17), and third is an isomorphism because $\eta _ X$ exhibits $F(X)$ as a $\operatorname{\mathcal{E}}_1$-reflection of $X$. $\square$