Remark 5.33.
On classes of small generation, the assignments \(\Sigma \mapsto \Sigma^m\) and \(\Sigma \mapsto \Sigma^c\) define the respective left adjoints to the inclusions of modalities and congruences into saturated classes. Without a smallness restriction, the same closure operations are left adjoint to the inclusions \(\Acyc(T)\hookrightarrow\Sat(T)\) and \(\Cong(T)\hookrightarrow\Sat(T)\) at the level of partially ordered classes.