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  "body": "\n## Summary\n\nThe 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.\nIt is the [[product|S5744QZV4N]] of $\\mathcal{C}$ and $\\mathcal{D}$ in the category of small categories.\nIt 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.\n\n## Explanations\n\n### L1 human\n\nThe product category is to categories what the [[Cartesian product|09CEDJYBKP]] is to sets.\nAn object is a pair, one object from each category, and an arrow is a pair of arrows, one in each category.\nTwo pairs of arrows chain by chaining each coordinate separately.\nFor 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 $b \\le b'$.\n",
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