# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 89: Functors

January 10, 2017

Posted by on In category theory, a **functor** is a map between categories consisting of two components:

- a map from the objects of to those of
- a map from each morphism in to a morphism in .

This map must preserve the identity morphisms and composition. That is, it must satisfy the following properties:

- for objects in

Functors can be thought of as kinds of homomorphisms between categories. With functors as arrows, categories can then form a category called , the category of categories.

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