Profunctor Equipment Implementation in Haskell

The author presents a toy implementation of profunctor equipment in Haskell, aiming to make the concepts verifiable by the compiler. The core of the implementation revolves around the Cell type, defined as a natural transformation between profunctors. Horizontal composition is handled by the hcomp function, while vertical composition uses vcomp, both leveraging the Procompose type for coends. The article also introduces helper types such as Hunit and Vunit, along with companion and conjoint types (Costar and Star). Unit laws are satisfied up to isomorphism, and the implementation references the proarrows library for more elaborate versions. The code demonstrates how to compose endo‑functors and endo‑profunctors within the Haskell category of types and functions. The author encourages readers to experiment with the provided functions to gain intuition about profunctor composition.

The implementation showcases the power of Haskell’s type system to encode complex categorical concepts, making the verification process more reliable than relying solely on AI tools. It also highlights the importance of explicit type definitions and natural transformations in functional programming.

The article concludes by pointing to the proarrows library for a full‑blown dependently‑typed version and encourages readers to explore further applications of profunctor equipment.