  The flow of information through a complex system can be readily understood
with category theory. However, negative information (e.g., what is not
possible) does not have an immediately evident categorical representation. The
formalization of nategories using unconventional composition addresses this
issue, and lets imposed limitations on categories be considered. However,
traditional nategories abandon core categorical constructs and rely on
extensive mathematical development. This creates a divide between the
consideration of positive and negative information composition. In this work,
we show that negative information can be considered in a natural categorical
manner. This is aided by functor string diagrams, a novel flexible diagrammatic
approach that can intuitively show the operation of hom-functors and natural
transformations in expressions. This insight reveals how to consider the
composition of negative information with foundational categorical constructs
without relying on enrichment. We present diagrammatic means to consider not
only nategories, but preorders more broadly. This paper introduces diagrammatic
methods for the consideration of triangle inequalities and co-designs
$\mathbf{DP/Feas_{Bool}}$, showing how important cases of negative information
composition can be categorically and diagrammatically approached. In
particular, we develop systematic tools to rigorously consider imposed
limitations on systems, advancing our mathematical understanding, and present
intuitive diagrams which motivate widespread adoption and usage for various
applications.
