equivalences in/of $(\infty,1)$-categories
The analog in (∞,1)-category theory of the/a adjoint functor theorem in ordinary category theory.
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.
adjoint $(\infty,1)$-functor theorem
Section 5.5 of