On unification of categories associated with F -transforms and fuzzy pretopological spaces as Qua category

In this paper, we develop a unified categorical framework for structures associated with F-transforms and fuzzy pretopological spaces. Unlike existing approaches based on function-induced morphisms, we introduce categories whose morphisms are defined as pairs of Formula: see text-valued fuzzy relations, allowing a more general and flexible representation of transformations. Specifically, we construct the categories of spaces with Formula: see text-valued fuzzy partitions, Formula: see text-valued fuzzy lower transformation systems, Formula: see text-valued fuzzy pretopological spaces, and Čech Formula: see text-valued fuzzy interior spaces, and establish isomorphisms and functorial relationships among them. We further prove the existence of adjoint functors between these categories. A central contribution of this work is to demonstrate that these categories naturally embed into the category Qua, thereby providing a common relational framework that captures their structural similarities. This unification not only generalizes existing categorical constructions but also reveals deeper connections between fuzzy relational systems and transformation-based models.