Kerodon

Proof. We first prove necessity. Suppose that there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which exhibits $F$ as a left adjoint of $G$. Fix an object $X \in \operatorname{\mathcal{C}}$ and set $Y = F(X)$. Then $\eta $ determines a morphism $\eta _{X}: X \rightarrow G(Y)$ which satisfies the requirement of condition $(\ast _ X)$ (Proposition 6.2.1.17).

We now prove sufficiency. Let $\operatorname{\mathcal{E}}$ denote the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be the cartesian fibration of Proposition 5.2.3.15. Let us abuse notation by identifying the fibers $\{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ with $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. Fix an object $X \in \operatorname{\mathcal{C}}$, and suppose that there exists an object $Y \in \operatorname{\mathcal{D}}$ together with a morphism $u: X \rightarrow G(Y)$ satisfying the requirement of condition $(\ast _ X)$. Then we can identify $u$ with a morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Our assumption on $u$ guarantees that the morphism $f$ is $U$-cocartesian (see Corollary 5.1.2.3). Consequently, if condition $(\ast _ X)$ is satisfied for every object $X \in \operatorname{\mathcal{C}}$, then $U$ is a cocartesian fibration. Applying Proposition 6.2.3.4, we conclude that $G$ admits a left adjoint. $\square$

Proof. The equivalence $(1) \Leftrightarrow (4)$ is a reformulation of Proposition 6.2.4.1. The implication $(2) \Rightarrow (3)$ is immediate. To see that $(3)$ implies $(4)$, we observe that if $\lambda : \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor which is corepresentable by an object $X \in \operatorname{\mathcal{C}}$, then $\lambda \circ \mathrm{h} \mathit{G}$ is isomorphic to the enriched homotopy transport representation of the left fibration $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{D}}$. If $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ has an initial object, then this functor is corepresentable by virtue of Proposition 5.6.6.21.

The implication $(4) \Rightarrow (5)$ follows from Remark 5.6.6.11. We will complete the proof by showing that $(5)$ implies $(2)$. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a left fibration, and let $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$ (see Definition 5.6.5.1). If $\widetilde{\operatorname{\mathcal{C}}}$ has an initial object, then the functor $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}}$ is corepresentable (Proposition 5.6.6.21). Assumption $(5)$ then guarantees that the functor $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}} \circ G$ is also corepresentable. Identifying $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}} \circ G$ with the covariant transport representation of the left fibration $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$, we see that the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$ also has an initial object (Proposition 5.6.6.21). $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.1.17, the implication $(2) \Rightarrow (3)$ follows from Proposition 6.2.4.1. We will complete the proof by showing that $(3) \Rightarrow (1)$. Note that, if condition $(3)$ is satisfied, then the natural transformation $\eta $ exhibits $\mathrm{h} \mathit{G}: \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a right adjoint of the functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ (see Variant 6.1.2.11). Invoking Proposition 6.2.1.14, we deduce that $\eta $ is the unit of an adjunction between $F$ and $G$. $\square$

Proof. We will use the criterion of Corollary 6.2.4.2. Fix an object $\overline{X} \in \operatorname{\mathcal{C}}_{ / (G \circ u) }$; we wish to show that the $\infty $-category

has an initial object. Let $X$ denote the image of $\overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we can identify $\overline{X}$ with a morphism of simplicial sets $\overline{u}: K \rightarrow (\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/})$, and $\operatorname{\mathcal{E}}$ with the slice $\infty $-category $(\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/})_{ / \overline{u} }$. Since $G$ admits a left adjoint, the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ has an initial object (Corollary 6.2.4.2). Applying Corollary 7.1.3.20, we conclude that $\operatorname{\mathcal{E}}$ also has an initial object. $\square$