S M Nazmuz Sakib's Formula for Adaptive Compositionality in Categories: A Context-Sensitive Approach to Functorial Composition

This paper studies S. M. Nazmuz Sakib’s Formula for Adaptive Compositionality in Categories as a context-sensitive extension of functorial composition. The main objective is to examine how a contextual factor can modify the composition of morphisms while preserving the fundamental categorical properties of associativity and identity. The research is conducted using a theoretical mathematics approach based on formal definition, structural analysis, and proof-oriented verification within category theory. The results show that the proposed contextual factor must satisfy the coherence conditions for the modified composition to remain mathematically consistent. In particular, the analysis indicates that identity preservation requires normalization conditions, while associativity requires compatibility across triples of composable morphisms. These findings clarify the mathematical conditions under which the formula can be treated as a valid categorical framework and provide a basis for further applications in topology, algebra, and dynamical systems