# Functoriality in Category Theory: Mappings That Preserve Morphisms

DevFeed: [Functoriality in Category Theory: Mappings That Preserve Morphisms](<https://devfeed.tech/articles/functoriality-40326.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/07/14/functoriality/>)

Published: 2013-07-14T10:03:29Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Category Theory](<https://devfeed.tech/topics/category-theory.md>), [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [categories](<https://devfeed.tech/tags/categories.md>), [category-theory](<https://devfeed.tech/tags/category-theory.md>), [coproducts](<https://devfeed.tech/tags/coproducts.md>), [functor](<https://devfeed.tech/tags/functor.md>), [homology](<https://devfeed.tech/tags/homology.md>), [ml](<https://devfeed.tech/tags/ml.md>), [morphisms](<https://devfeed.tech/tags/morphisms.md>), [universal-properties](<https://devfeed.tech/tags/universal-properties.md>)

## AI overview

This tutorial introduces functoriality in category theory. It explains functors as mappings between categories that assign objects and morphisms while preserving identity morphisms and composition, with homology as an example of a functorial construction.

## Source excerpt

Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted at: functoriality. As we've repeatedly stressed, the meat of category theory is in the morphisms. One natural question one might ask is, what notion of morphism is there between categories themselves?