# Profunctor Equipment

A double-category structure in which every vertical arrow has a horizontal companion and every companion has a conjoint, used to define categorical constructions such as weighted limits and pointwise Kan extensions; it also has a toy Haskell implementation.

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## Bending, Yanking, and Cartesian Squares in Double Categories

DevFeed: [Bending, Yanking, and Cartesian Squares in Double Categories](<https://devfeed.tech/articles/bending-yanking-and-cartesian-squares-in-double-categories-28857.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/05/18/bending-yanking-and-cartesian-squares-in-double-categories/>)

Author: Bartosz Milewski

Published: 2026-05-19T06:54:15Z

Content type: article

Language: en

Sources: [Bartosz Milewski's Programming Cafe](<https://devfeed.tech/sources/bartosz-milewski-s-programming-cafe.md>)

Topics: [Category Theory](<https://devfeed.tech/topics/category-theory.md>), [Profunctor Equipment](<https://devfeed.tech/topics/profunctor-equipment.md>), [String Diagrams](<https://devfeed.tech/topics/string-diagrams.md>)

Tags: [category-theory](<https://devfeed.tech/tags/category-theory.md>), [diagram](<https://devfeed.tech/tags/diagram.md>), [double-categories](<https://devfeed.tech/tags/double-categories.md>), [proarrow-equipment](<https://devfeed.tech/tags/proarrow-equipment.md>), [profunctor-equipment](<https://devfeed.tech/tags/profunctor-equipment.md>), [profunctors](<https://devfeed.tech/tags/profunctors.md>), [string-diagrams](<https://devfeed.tech/tags/string-diagrams.md>)

### AI overview

This article explains how string-diagram manipulations in double categories and proarrow equipment support yanking identities and the spider lemma, then introduces Cartesian squares as a universal construction in category theory.

### Source excerpt

Previously: Profunctor Equipment in Haskell. The major advantage of string diagrams is that they provide surprisingly natural language for complex diagram manipulations. The fact that two traditional diagrams are equal can be often described as a permission to bend, yank, or pinch strings in particular ways. They provide visual and often tactile clues to our [...]