# Kan Extensions in Double Categories

DevFeed: [Kan Extensions in Double Categories](<https://devfeed.tech/articles/kan-extensions-in-double-categories-28860.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/06/13/kan-extensions-in-double-categories/>)

Author: Bartosz Milewski

Published: 2026-06-13T12:27:28Z

Content type: article

Language: en

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

Topics: [Haskell](<https://devfeed.tech/topics/haskell.md>), [data type](<https://devfeed.tech/topics/data-type.md>), [implementation](<https://devfeed.tech/topics/implementation.md>)

Tags: [category-theory](<https://devfeed.tech/tags/category-theory.md>), [data-type](<https://devfeed.tech/tags/data-type.md>), [double-category](<https://devfeed.tech/tags/double-category.md>), [function](<https://devfeed.tech/tags/function.md>), [haskell](<https://devfeed.tech/tags/haskell.md>), [implementation](<https://devfeed.tech/tags/implementation.md>), [kan-extensions](<https://devfeed.tech/tags/kan-extensions.md>), [profunctor-equipment](<https://devfeed.tech/tags/profunctor-equipment.md>), [profunctors](<https://devfeed.tech/tags/profunctors.md>)

## AI overview

This article generalizes right and left Kan extensions from functors to profunctors in double categories and presents corresponding Haskell representations. It explains the associated universal and factorization properties, including their computational interpretation.

## Source excerpt

Previously: Kan extensions in Haskell. In a double category that is also a proarrow equipment, we have the ability to bend arrows. In particular, in the definition of the counit of the right Kan extension: we can bend the vertical arrow, replacing it with its horizontal conjoint . In a profunctor equipment, this is just [...]