Category theory provides a general mathematical framework for comparing formal structures. We have been using category theory towards an understanding of so-called dual-process cognition: the view that cognition is composed of competing and complementary styles of thinking: e.g., fast and slow, reflexive and reflective, intuitive and rational, and so on. In this regard, I will present some empirical studies and their category theoretical motivation/justification. I will also discuss potential ways in which category theory can further contribute to an understanding of cognition