Profunctor Equipment

This article delves into advanced category theory concepts, focusing on profunctors as a way to capture hom-set information without peeking inside categories. It discusses how categories, functors, and natural transformations form a 2-category, and how profunctors extend this to a bicategory. The text explains the composition of profunctors via coends and introduces a double category that combines vertical functors with horizontal profunctors, using 2-cells to relate them. The content is highly technical and relevant to computer science, particularly in areas like functional programming, type theory, and categorical semantics.