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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be [[categories|A2DE66937D]].\nThe *product category* $\\mathcal{C} \\times \\mathcal{D}$ is defined as follows:\n\n1. its objects are the pairs $(A, B)$ with $A$ an object of $\\mathcal{C}$ and $B$ an object of $\\mathcal{D}$;\n2. a [[morphism|859F0TFVA3]] $(A, B) \\to (A', B')$ is a pair $(f, g)$ with $f \\colon A \\to A'$ in $\\mathcal{C}$ and $g \\colon B \\to B'$ in $\\mathcal{D}$;\n3. composition is $(f', g') \\circ (f, g) = (f' \\circ f, g' \\circ g)$, and the identity of $(A, B)$ is $(1_A, 1_B)$.\n\n## Remarks\n\nThe category axioms hold because they hold in each coordinate.\nThere are projection functors $\\mathcal{C} \\times \\mathcal{D} \\to \\mathcal{C}$ and $\\mathcal{C} \\times \\mathcal{D} \\to \\mathcal{D}$.\nExample: a [[monoid|0N6R902FG1]] $M$ is a category with one object; the product of the one-object categories of $M$ and $N$ is the one-object category of the product monoid $M \\times N$.\nIndeed its only object is the pair of the two objects, and its morphisms are the pairs $(m, n)$ multiplied coordinatewise.\nThe set of objects of $\\mathcal{C} \\times \\mathcal{D}$ is the [[Cartesian product|09CEDJYBKP]] of the sets of objects when both categories are small.\nA morphism $(f, g)$ is an isomorphism exactly when $f$ and $g$ are both isomorphisms.\n",
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