> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/10-functors/adjoint-functor.md). # Adjoint functor **Adjunction** is a relationship that two functors may have. Two functors that stand in adjunction are known as **adjoint functors**, one being the *left adjoint* and the other the *right adjoint*. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e. constructions of objects having a certain universal property). By definition, an *adjunction* between categories `C` and `D` is a pair of functors (assumed to be covariant) `F : C → D` and `G : D → C` and, for all objects `X` in `C` and `Y` in `D`, a bijection between the respective morphism sets `Homᴄ(FY, X) ≅ Homᴅ(Y, GX)` such that this family of bijections is *natural* in `X` and `Y`. Categorical Programming With Inductive And Coinductive Types --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/450-category-theory/10-functors/adjoint-functor.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.