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  "body": "\n## Definition\n\nLet $\\Sigma$ be a statement in the first-order language of [[categories|A2DE66937D]], built from objects, morphisms, domains, codomains, identities and composition.\nThe *dual statement* $\\Sigma^{\\mathrm{op}}$ is obtained from $\\Sigma$ by these changes:\n\n1. each domain is replaced by the codomain, and each codomain by the domain;\n2. each composite $g \\circ f$ is replaced by $f \\circ g$.\n\nThe *duality principle* states that $\\Sigma$ holds in the [[opposite category|DAMK8XE1FR]] $\\mathcal{C}^{\\mathrm{op}}$ if and only if $\\Sigma^{\\mathrm{op}}$ holds in $\\mathcal{C}$.\nIn particular, if $\\Sigma$ holds in every category, then so does $\\Sigma^{\\mathrm{op}}$.\n\n## Remarks\n\nTwo notions are *dual* when the defining statement of one is the dual of the defining statement of the other.\nExample: \"$f$ is a [[monomorphism|Y2NP1JFR86]]\" says that $f \\circ g_1 = f \\circ g_2$ implies $g_1 = g_2$.\nIts dual says that $g_1 \\circ f = g_2 \\circ f$ implies $g_1 = g_2$, which is \"$f$ is an [[epimorphism|RVXDSY04KG]]\".\nHence the fact \"a composite of monomorphisms is a monomorphism\" gives at once \"a composite of epimorphisms is an epimorphism\".\nLikewise a section and a retraction are dual, as in [[section and retraction|DXJH3DGXBC]].\nNon-example: the dual of a statement true in one particular category need not be true in that category.\nIn the category of sets, every morphism $X \\to \\emptyset$ is an isomorphism, since $X$ must then be empty.\nThe dual statement says that every morphism $\\set{0} \\to X$ out of the one-point set is an isomorphism, and it fails for $X = \\set{0, 1}$.\nDuality in this sense is different from the duality of vector spaces, although the [[dual space|07JBSZE0V4]] functor turns linear maps around.\n",
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