Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Recall that, if $F$ admits a right adjoint, then it preserves all colimit diagrams which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.1.4.28). Similarly, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a functor which admits a left adjoint, then it preserves all limit diagrams which exist in $\operatorname{\mathcal{D}}$. In the setting of presentable $\infty $-categories, we have the following converse:
Theorem 9.5.2.1 (Adjoint Functor Theorem). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then:
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it preserves small colimits.
A functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ admits a left adjoint if and only if it is accessible and preserves small limits.
The first half of Theorem 9.5.2.1 is a consequence of the following more general result:
Proposition 9.5.2.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is presentable and $\operatorname{\mathcal{D}}$ is locally small. Then $F$ admits a right adjoint if and only if it is cocontinuous: that is, it preserves small colimits.
Proof. Assume that $F$ is cocontinuous; we will show that it admits a right adjoint (the reverse implication follows from Corollary 7.1.4.28). By virtue of Corollary 6.2.6.2, it will suffice to show that for every functor $h_{D}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ which is representable by an object $D \in \operatorname{\mathcal{D}}$, the composition $(h_{D} \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by an object of $\operatorname{\mathcal{C}}$. Using Theorem 9.5.1.8, we are reduced to showing that the functor $h_{D} \circ F^{\operatorname{op}}$ preserves small limits. This follows from the cocontinuity of $F$, since $h_{D}$ preserves all limits which exist in $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (Corollary 7.4.1.23). $\square$
The second half of Theorem 9.5.2.1 is a consequence of the following:
Proposition 9.5.2.3. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is accessible and $\operatorname{\mathcal{D}}$ is presentable. Then $G$ admits a left adjoint if and only if it is accessible and preserves small limits.
Beware that the accessibility hypothesis in Proposition 9.5.2.3 cannot be omitted (see Warning 9.5.1.10).
Proof of Proposition 9.5.2.3. Assume that $G$ is accessible and preserves small limits; we wish to show that $G$ admits a left adjoint (the converse follows from Corollaries 7.1.4.28 and 9.4.7.18). By virtue of Corollary 6.2.6.2, it will suffice to show that for every functor $h_ C: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by an object $C \in \operatorname{\mathcal{C}}$, the composition $(h_{C} \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by an object of $\operatorname{\mathcal{D}}$. Using the criterion of Theorem 9.5.1.9, we are reduced to showing that $h_{C} \circ G$ is accessible and preserves small limits. This follows from our hypotheses on $G$, since the functor $h_{C}$ is accessible (Example 9.4.7.16) and preserves all limits which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.4.1.23). $\square$
Proof of Theorem 9.5.2.1. Combine Propositions 9.5.2.2 and 9.5.2.3. $\square$
Remark 9.5.2.4. The preceding arguments show that Propositions 9.5.2.2 and 9.5.2.3 are formal consequences of Theorems 9.5.1.8 and 9.5.1.9 (respectively). The reverse is also true: assuming Propositions 9.5.2.2 and 9.5.2.3, we can immediately deduce the characterizations of representable and corepresentable functors supplied by Theorems 9.5.1.8 and 9.5.1.9. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Then:
If $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is a continuous functor, then Proposition 9.5.2.2 guarantees that the opposite functor $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{\operatorname{op}}$ admits a right adjoint $T: \operatorname{\mathcal{S}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. It is then easy to see that the functor $\mathscr {F}$ is representable by the object $T( \Delta ^0 ) \in \operatorname{\mathcal{C}}$.
If $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a continuous accessible functor, then Proposition 9.5.2.3 guarantees that $\mathscr {G}$ admits a left adjoint $U: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{C}}$. It is then easy to see that the functor $\mathscr {G}$ is corepresentable by the object $U( \Delta ^0 ) \in \operatorname{\mathcal{C}}$.
We close this section by recording more general forms of Propositions 9.5.2.2 and 9.5.2.3, which do not require any inaccessible cardinals.
Variant 9.5.2.5. Let $\kappa < \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $(\kappa ,\lambda )$-compactly generated, and suppose that the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Then a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it satisfies the following pair of conditions:
The functor $F$ is $\lambda $-cocontinuous.
For every object $C \in \operatorname{\mathcal{C}}_{< \kappa }$ and every object $D \in \operatorname{\mathcal{D}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D )$ is essentially $\lambda $-small.
Proof. Fix an uncountable regular cardinal $\mu $ such that $\operatorname{\mathcal{D}}$ is locally $\mu $-small, so that every object $D \in \operatorname{\mathcal{D}}$ determines a representable functor $h_{D}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$. By virtue of Corollary 6.2.6.2, the functor $F$ admits a right adjoint if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the composition $\operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow {F^{\operatorname{op}}} \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { h_{D} } \operatorname{\mathcal{S}}_{< \mu }$ is representable by an object of $\operatorname{\mathcal{C}}$. The equivalence of this condition with $(1)$ and $(2)$ now follows from Proposition 9.5.1.11 (together with Proposition 7.4.1.22). $\square$
Variant 9.5.2.6. Let $\kappa \leq \lambda \leq \mu $ be regular cardinals, where $\lambda $ is uncountable and $\mu $ has exponential cofinality $\geq \lambda $. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $(\kappa ,\mu )$-compactly generated and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor which is $(\kappa ,\mu )$-finitary. Assume that $\operatorname{\mathcal{D}}$ is $\lambda $-complete and that the full subcategory $\operatorname{\mathcal{D}}_{< \kappa } \subseteq \operatorname{\mathcal{D}}$ spanned by the $(\kappa ,\mu )$-compact objects is essentially $\lambda $-small. Then $G$ admits a left adjoint if and only if it satisfies the following conditions:
The functor $G$ is $\lambda $-continuous: that is, it preserves $\lambda $-small limits.
For every pair of objects $C \in \operatorname{\mathcal{C}}_{< \kappa }$ and $D \in \operatorname{\mathcal{D}}_{< \kappa }$, the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, G(D) )$ is essentially $\lambda $-small.
Proof. Assume first that $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. In this case, condition $(1)$ follows from Corollary 7.1.4.28. To prove $(2)$, note that our assumption that $G$ is $(\kappa ,\mu )$-finitary guarantees that $F$ is $(\kappa ,\mu )$-compact (Example 9.4.2.17). In particular, if $C \in \operatorname{\mathcal{C}}_{< \kappa }$, then $F(C) \in \operatorname{\mathcal{D}}_{< \kappa }$, so that the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, G(D) ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D )$ is essentially $\lambda $-small for any $D \in \operatorname{\mathcal{D}}_{< \kappa }$ by virtue of our assumption that the $\infty $-category $\operatorname{\mathcal{D}}_{< \kappa }$ is essentially $\lambda $-small.
We now prove the converse. Assume that conditions $(1)$ and $(2)$ are satisfied; we wish to show that $G$ admits a left adjoint. Fix a regular cardinal $\nu $ of exponential cofinality $\geq \mu $ such that $\operatorname{\mathcal{C}}$ is locally $\nu $-small. For each object $C \in \operatorname{\mathcal{C}}$, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ be the functor corepresented by $C$. By virtue of Corollary 6.2.6.2, it will suffice to show that each of the functors $(h^{C} \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ is corepresentable by an object of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is $(\kappa ,\mu )$-cocomplete, the collection of objects $C \in \operatorname{\mathcal{C}}$ which satisfy this condition is closed under the formation of $\mu $-small $\kappa $-filtered colimits. We may therefore assume without loss of generality that the object $C \in \operatorname{\mathcal{C}}_{< \kappa }$ is $(\kappa ,\mu )$-compact. In this case, we will show that the functor $\mathscr {F} = h^{C} \circ G$ is corepresentable by a $(\kappa ,\mu )$-compact object of $\operatorname{\mathcal{D}}$. Since $C$ is $(\kappa ,\mu )$-compact and the functor $G$ is $(\kappa ,\mu )$-finitary, the functor $\mathscr {F}$ is also $(\kappa ,\mu )$-finitary. It follows from assumption $(2)$ that the space $\mathscr {F}(D) = \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, G(D) )$ is essentially $\lambda $-small for $D \in \operatorname{\mathcal{D}}_{< \kappa }$, and is therefore essentially $\mu $-small for any object $D \in \operatorname{\mathcal{D}}$. Invoking the criterion of Proposition 9.5.1.13, we are reduced to showing that the functor $\mathscr {F}$ is $\lambda $-continuous. This follows from assumption $(1)$, since the corepresentable functor $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ preserves all limits which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.4.1.23). $\square$