---
tag: AFNE9C4NH6
kind: concept
name: Product category
msc: [18A05]
sources:
  - url: https://en.wikipedia.org/wiki/Product_category
same_as:
  wikidata: Q13443840
  wikipedia: Product category
---

## Summary

The product category $\mathcal{C} \times \mathcal{D}$ of two [[categories|A2DE66937D]] has pairs of objects as objects and pairs of morphisms as morphisms, composed in each coordinate.
It is the [[product|S5744QZV4N]] of $\mathcal{C}$ and $\mathcal{D}$ in the category of small categories.
It is needed to speak of [[functors|28NQ0KRHD3]] of two variables, such as the hom functor $\Hom(-, -)$ on $\mathcal{C}^{\mathrm{op}} \times \mathcal{C}$ or the tensor product of modules.

## Explanations

### L1 human

The product category is to categories what the [[Cartesian product|09CEDJYBKP]] is to sets.
An object is a pair, one object from each category, and an arrow is a pair of arrows, one in each category.
Two pairs of arrows chain by chaining each coordinate separately.
For example, if $\mathcal{C}$ and $\mathcal{D}$ are both the order $0 \le 1$, then $\mathcal{C} \times \mathcal{D}$ is the order on the four pairs $(a, b)$ with $(a, b) \le (a', b')$ when $a \le a'$ and $b \le b'$.