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  "body": "\n## Summary\n\nThe product of a family of objects in a [[category|A2DE66937D]] is an object with a projection to each member of the family.\nAny family of morphisms into the members factors uniquely through it.\nIt is the [[limit|CHS5AFJGWE]] of a diagram with no non-identity arrows.\nIt generalises the [[Cartesian product|09CEDJYBKP]] of sets, the [[direct product of groups|00HF4RJ5QN]] and the product of topological spaces.\n\n## Explanations\n\n### L1 human\n\nA map into a product is the same thing as a list of maps, one into each factor.\nFor sets, a function $f \\colon Z \\to X \\times Y$ is the same as the pair of functions $z \\mapsto$ first coordinate and $z \\mapsto$ second coordinate of $f(z)$.\nThe categorical product takes this property as the definition and forgets how the product is built.\nIn a partially ordered set seen as a category, the product of two elements is their greatest lower bound.\n",
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