equivalences in/of $(\infty,1)$-categories
The analogs in (∞,1)-category theory of the adjoint functor theorems in ordinary category theory.
Let $G:\mathcal{D} \to \mathcal{C}$ be a functor between $(\infty, 1)$-categories. We say that $G$ satisfies the solution set condition if the $(\infty, 1)$-category $c/G$ admits a small weakly initial set for any $c$ in $\mathcal{C}$.
$G$ satisfies the $h$-initial object condition if $c/G$ admits an h-initial object for any $c$ in $\mathcal{C}$.
$\mathcal{C}$ is said to be 2-locally small if for every pair of objects, $x$,$y$, of $\mathcal{C}$, the mapping space $map_{\mathcal{C}}(x,y)$ is locally small.
Let $G:\mathcal{D} \to \mathcal{C}$ be a continuous functor. Suppose that $\mathcal{D}$ is locally small and complete and $\mathcal{C}$ is 2-locally small. Then $G$ admits a left adjoint if and only if it satisfies the solution set condition.
Let $G:\mathcal{D} \to \mathcal{C}$ be a finitely continuous functor?. Suppose that $\mathcal{D}$ is finitely complete. Then $G$ admits a left adjoint if and only if it satisfies the $h$-initial object condition.
See Section 3 of (NRS18).
The following result is a consequence.
Let $F : C \to D$ be an (∞,1)-functor between locally presentable (∞,1)-categories then
it has a right adjoint (∞,1)-functor precisely if it preserves small colimits;
it has a left adjoint (∞,1)-functor precisely if it is an accessible (∞,1)-functor and preserves small limits.
This is HTT, cor. 5.5.2.9.
For the existence of right adjoints, we can weaken the hypotheses to merely requiring $D$ to be a locally small (infinity,1)-category. (HTT, rem. 5.5.2.10)
Section 5.5 of
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