More on injectivity in locally presentable categories

Injectivity with respect to morphisms having k-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, k-directed colimits and k-pure subobjects. This sharpens the result of the first two authors. In contrast, for geometric logic an example is found of a class closed under directed colimits and pure subobjects, but not axiomatizable by a geometric theory.