We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. Such (...) categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with. (shrink)