---
title: Natural Transformations
description: Natural transformations provide a systematic way to compare two parallel functors. Learners will understand how to map one functor to another while preserving the underlying categorical structure, formalizing the concept of a natural isomorphism.
category: mathematics
subcategory: category-theory
difficulty: beginner, intermediate, advanced
url: /subject/natural-transformations
---

# Natural Transformations

Natural transformations provide a systematic way to compare two parallel functors. Learners will understand how to map one functor to another while preserving the underlying categorical structure, formalizing the concept of a natural isomorphism.

## Where to start

Start with Category Theory for Programmers, Bartosz Milewski's free video course, which approaches natural transformations from a functional programmer's point of view. It is also the one to use if you only use one resource, since the other entries are reference material. For precise definitions, examples and references, move on to nLab: Natural Transformation.

## Available Resources

1 Courses • 3 Websites

## Websites

### 1. nLab: Natural Transformation

The nLab wiki entry defining natural transformations between functors, with component and naturality-square formulations, vertical and horizontal composition, and links to 2-categorical generalizations. Best used as a precise reference once the basic definitions of category and functor are familiar.

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

**Link:** https://ncatlab.org/nlab/show/natural+transformation

**Tags:** category-theory, natural-transformations, functors, 2-categories

### 2. 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

### 3. 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

## Courses

### 1. Category Theory for Programmers

Learn natural transformations in Category Theory for Programmers with Bartosz Milewski's insightful video course. Master functional programming concepts!

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

**Link:** https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_

**Tags:** category-theory, natural-transformations, functors, functional-programming, haskell

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