# Proarrow Equipment

Published articles for Proarrow Equipment.

This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles.

## Yoneda Lemma in Double Categories

DevFeed: [Yoneda Lemma in Double Categories](<https://devfeed.tech/articles/yoneda-lemma-in-double-categories-28865.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/09/13/yoneda-lemma-in-double-categories/>)

Author: Bartosz Milewski

Published: 2026-09-13T12:23:31Z

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>), [Profunctors](<https://devfeed.tech/topics/profunctors.md>), [String Diagrams](<https://devfeed.tech/topics/string-diagrams.md>), [Haskell](<https://devfeed.tech/topics/haskell.md>)

Tags: [category-theory](<https://devfeed.tech/tags/category-theory.md>), [double-category](<https://devfeed.tech/tags/double-category.md>), [implementation](<https://devfeed.tech/tags/implementation.md>), [kan-extensions](<https://devfeed.tech/tags/kan-extensions.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>), [yoneda-structure](<https://devfeed.tech/tags/yoneda-structure.md>)

### AI overview

The article explains how to formulate the Yoneda lemma in double categories without directly referring to presheaves or hom-sets. It uses profunctors, universal constructions, Kan extensions, tabulations, 2-cells, and string diagrams to describe the Yoneda embedding and its desired properties, including density and full faithfulness.

### Source excerpt

Working with double categories can be aptly summarized in a meme: Talk to me about sets without mentioning sets. We don't talk about hom-sets, we talk about horizontal units. Secretly, we are visualizing horizontal arrows as profunctors, and the unit of profunctor composition is a hom-functor. Presheaves are defined as -valued functors, so we immediately [...]

## Tambara Equipment

DevFeed: [Tambara Equipment](<https://devfeed.tech/articles/tambara-equipment-28862.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/07/11/tambara-equipment/>)

Author: Bartosz Milewski

Published: 2026-07-11T08:13:34Z

Content type: article

Language: en

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

Topics: [modules](<https://devfeed.tech/topics/modules.md>), [Haskell](<https://devfeed.tech/topics/haskell.md>), [Code](<https://devfeed.tech/topics/code.md>)

Tags: [category-theory](<https://devfeed.tech/tags/category-theory.md>), [code](<https://devfeed.tech/tags/code.md>), [double-categories](<https://devfeed.tech/tags/double-categories.md>), [double-category](<https://devfeed.tech/tags/double-category.md>), [haskell](<https://devfeed.tech/tags/haskell.md>), [modules](<https://devfeed.tech/tags/modules.md>), [optics](<https://devfeed.tech/tags/optics.md>), [proarrow-equipment](<https://devfeed.tech/tags/proarrow-equipment.md>), [profunctors](<https://devfeed.tech/tags/profunctors.md>), [structure](<https://devfeed.tech/tags/structure.md>), [tambara-modules](<https://devfeed.tech/tags/tambara-modules.md>), [tannakian-reconstruction](<https://devfeed.tech/tags/tannakian-reconstruction.md>), [theory](<https://devfeed.tech/tags/theory.md>), [transformation](<https://devfeed.tech/tags/transformation.md>)

### AI overview

This article explains Tambara modules through category theory and illustrates the concepts with Haskell code. It discusses their relationship to profunctors, monoidal actions, double categories, proarrow equipment, and Tannakian reconstruction.

### Source excerpt

I was originally attracted to category theory when trying to understand Haskell optics. I was puzzled by the van Laarhoven's functor representations and Kmett's use of Tambara modules. By playing Tetris with the Yoneda lemma I was able to make some progress, attacking more and more esoteric topics. With a group of researcher and students [...]

## 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 [...]