Proposition 7.7.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite limits, let $X$ be an object of $\operatorname{\mathcal{C}}$, let $F: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ be the forgetful functor, and let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{/X}$ be the right adjoint to $F$. The following conditions are equivalent:
- $(1)$
For every object $Y \in \operatorname{\mathcal{C}}$, there exists an exponential $Y^{X}$.
- $(2)$
The functor $G$ admits a right adjoint $H: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$.
Proof.
If condition $(2)$ is satisfied, then the composite functor $(F \circ G): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ also admits a right adjoint, given by the composition $(H \circ G): \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/X}$ (Remark 6.2.1.8). Consequently, the implication $(2) \Rightarrow (1)$ follows from Remark 7.7.3.12 (and does not require the assumption that $\operatorname{\mathcal{C}}$ admits finite limits). We now prove the converse. Assume that condition $(1)$ is satisfied; we wish to show that the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{/X}$ admits a right adjoint. Let $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}_{/X}$ denote the full subcategory spanned by those objects $\widetilde{Y}$ for which the functor $C \mapsto \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/X} }( G(C), \widetilde{Y} )$ is representable by an object of $\operatorname{\mathcal{C}}$. By virtue of Corollary 6.2.4.2, it will suffice to show that $\operatorname{\mathcal{D}}$ contains every object $\widetilde{Y}$ of $\operatorname{\mathcal{D}}$.
Let us identify $\widetilde{Y}$ with a morphism $u: Y \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $u$ lifts to a morphism $\widetilde{u}: \widetilde{Y} \rightarrow \widetilde{X}$ of $\operatorname{\mathcal{C}}_{/X}$, corresponding to the degenerate $2$-simplex
\[ \xymatrix { & X \ar [dr]^{\operatorname{id}} & \\ Y \ar [ur]^{u} \ar [rr]^{u} & & X } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $\eta : \operatorname{id}_{ \operatorname{\mathcal{C}}_{/X} } \rightarrow G \circ F$ and $\epsilon : (F \circ G) \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ be compatible unit and counit for an adjunction between $G$ and $F$. The we have a commutative diagram
7.84
\begin{equation} \begin{gathered}\label{equation:exponential-as-adjoint} \xymatrix { \widetilde{Y} \ar [r]^{\eta _{ \widetilde{Y}}} \ar [d]^{\widetilde{u}} & (G \circ F)( \widetilde{Y} ) \ar [d]^{ (G \circ F)(\widetilde{u})} \\ \widetilde{X} \ar [r]^{ \eta _{ \widetilde{X} }} & (G \circ F)(\widetilde{X} ) } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$. Note that the image of (7.84) under the functor $F$ can be identified with the left side of a the commutative diagram
\[ \xymatrix { Y \ar [d]^{u} \ar [r] & Y \times X \ar [r]^{ \epsilon _{ Y }} \ar [d] & Y \ar [d]^{u} \\ X \ar [r] & X \times X \ar [r] & X, } \]
where the left square and outer rectangle are pullback squares. It follows from Proposition 7.6.2.28 that the left square is also a pullback, so that (7.84) is a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ (Corollary 7.1.6.19). Our assumption that $\operatorname{\mathcal{C}}$ admits finite limits guarantees that the subcategory $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}_{/X}$ is closed under finite limits (Proposition 7.4.1.18). Consequently, to prove that $\widetilde{Y}$ belongs to $\operatorname{\mathcal{D}}$, it will suffice to show that the objects $\widetilde{X}$, $(G \circ F)( \widetilde{Y} )$, and $(G \circ F)( \widetilde{X})$ are contained in $\operatorname{\mathcal{D}}$. Note that each of these objects belongs to the essential image of the functor $G$ (the object $\widetilde{X}$ can obtained by applying $G$ to a final object of $\operatorname{\mathcal{C}}$). We are therefore reduced to proving that, for each object $Z \in \operatorname{\mathcal{C}}$, the functor
\[ C \mapsto \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/X} }( G(C), G(Z) ) \simeq \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( (F \circ G)(C), Z) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C \times X, Z) \]
is representable by an object of $\operatorname{\mathcal{C}}$, which is a reformulation of assumption $(1)$.
$\square$