The specific component of the applied aspect of the natural part of the IFF currently represents the area of first order logic and object level ontologies. Future areas covered may include semiotics, modal logic, fuzzy set theory, topic maps, etcetera. Currently, there are two meta-ontologies in the specific component — the IFF First Order Logic (meta) Ontology (IFF-FOL) and the IFF Ontology (meta) Ontology (IFF-ONT), both specific and non-parametric. There is only one copy of these meta-ontologies, and these are located at metalevel ‘sml’ = ‘1’. Both meta-ontologies represent numerous categories, functors, natural transformations, adjunctions and monads dealing with first order logic and object level ontologies. In the near future, these specific meta-ontologies will be intimately linked with (IFF-INS), the generic meta-ontology for institution theory. The IFF-FOL represents one-sorted traditional first order (categorical) logic and model theory. A many-sorted version is planned in the future. The IFF-OBJ represents a novel approach to first order logic and model theory. As an example of its special refinement, single relationships (tuples) can be regarded as full-blown models in its model theory. The concept of a polycosmic modularized object-level ontology has some interesting special cases in the IFF-OBJ.