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  "body": "\n## Summary\n\nDuality in category theory is the correspondence between a statement about a [[category|A2DE66937D]] and the statement obtained by reversing all arrows.\nA statement holds in the [[opposite category|DAMK8XE1FR]] $\\mathcal{C}^{\\mathrm{op}}$ exactly when its dual holds in $\\mathcal{C}$.\nSo a theorem proved for all categories gives, for free, its dual theorem.\nPairs of dual notions include [[monomorphism|Y2NP1JFR86]] and [[epimorphism|RVXDSY04KG]], [[initial and terminal objects|EK85VSSRMR]], [[product|S5744QZV4N]] and [[coproduct|4FJ0S71P58]], and [[limit|CHS5AFJGWE]] and [[colimit|BJS1CGTKQ5]].\n\n## Explanations\n\n### L1 human\n\nMany notions in category theory come in pairs that are mirror images of each other.\nTo pass from one to the other, turn every arrow around and read every composite in the opposite order.\nFor example, \"every arrow out of $X$ is unique\" becomes \"every arrow into $X$ is unique\".\nThe first describes an initial object, such as the empty set among sets; the second describes a terminal object, such as a one-point set.\nEach proof about one notion turns into a proof about its mirror image, so each theorem comes with a second, dual theorem at no extra cost.\n",
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