# Profunctors

A profunctor is a functor from the opposite of one category and another category to sets; profunctors are formalized in type theory and category-theoretic computer science.

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## 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 [...]