  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 $\infty$-categories -- in particular, 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 $\infty$-categories if $n \geq 2$ and the
homotopy $n$-categories of stable presentable $\infty$-categories for any $n
\geq 1$.
