Neuro-symbolic systems are a promising paradigm for Hybrid Intelligence (HI), combining the complementary strengths of symbolic reasoning and machine learning in human-AI interactions. However, designing architectures that satisfy desired properties currently relies on informal reasoning, rather than a rigorous framework. Our research addresses this gap by developing a formal intermediate layer that links architecture design to system properties. Building on the visual pattern language of the Boxology for neuro-symbolic systems, we develop a mathematical formalization that is grounded in Category Theory, in order to enable proof-based guarantees of properties. The soundness of this formalism will be demonstrated through empirical case studies of existing Hybrid Intelligence systems. This work advances the theoretical foundations of neuro-symbolic AI by supporting principled design of HI systems.