---
title: Adjoint Functors
description: Adjoint functors describe a fundamental relationship of duality between two categories. You will understand how to identify left and right adjoints, construct natural isomorphisms, and apply these relationships to simplify algebraic structures and preserve limits and colimits.
category: mathematics
subcategory: category-theory
difficulty: beginner, intermediate, advanced
url: /subject/adjoint-functors
---

# Adjoint Functors

Adjoint functors describe a fundamental relationship of duality between two categories. You will understand how to identify left and right adjoints, construct natural isomorphisms, and apply these relationships to simplify algebraic structures and preserve limits and colimits.

## Available Resources

4 Websites

## Websites

### 1. nLab: Adjoint Functor

Community-maintained research wiki entry on adjunctions, giving the hom-set, unit-counit and universal-arrow definitions with many examples and properties. Useful for checking precise statements and finding references once the basic definition is familiar from a textbook.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/adjoint+functor

**Tags:** category-theory, adjoint-functors, adjunctions, universal-properties, reference-wiki

### 2. What is an Adjunction? (Parts 1–3)

**Author:** Tai-Danae Bradley

Three-part blog series that builds intuition for adjunctions from the free-forgetful pair between sets and vector spaces. Later parts present the hom-set definition and the unit and counit definition and show they are equivalent. The best free bridge between seeing the definition and understanding why it is natural, for learners who find the textbooks too dense.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.math3ma.com/blog/what-is-an-adjunction-part-1

**Tags:** category-theory, adjunctions, free-forgetful-adjunction, hom-set-adjunction, unit-counit

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 4. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

## Youtubes

### 1. The Catsters: Adjunctions

**Author:** Eugenia Cheng, Simon Willerton

Short chalkboard lecture series by Eugenia Cheng and Simon Willerton. It defines adjunctions through unit, counit and the triangle identities and shows equivalence with the hom-set definition. Follow-on videos cover adjunctions, monads and universal arrows.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.youtube.com/watch?v=loOJxIOmShE

**Tags:** category-theory, adjunctions, unit-counit, triangle-identities, monads

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