Higher weak (co)limits, adjoint functor theorems, and higher Brown representability

We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\mathrm{\infty }$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\mathrm{\infty }$-categories if $n\ge 2$, and the homotopy $n$-categories of presentable stable $\mathrm{\infty }$-categories for any $n \geq 1$.