Pullback in the Category of Intuitionistic Fuzzy Modules

Pullbacks are fundamental limit constructions in category theory. Their study within intuitionistic fuzzy module categories, which provide a framework for handling mathematical uncertainty, remains underdeveloped. This work establishes their theoretical foundation in such categories. We first investigate the explicit construction of pullbacks and then study the properties of double pullbacks, monomorphisms, and epimorphisms. A classification of different pullback categories is provided. Key results include characterising pullback construction conditions, describing features of double pullbacks and related morphisms, and rigorously establishing the relationship between pullbacks and exactness. This work deepens the understanding of limit theory in intuitionistic fuzzy settings and provides a foundation for future research.