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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nThe *opposite category* $\\mathcal{C}^{\\mathrm{op}}$ is the category defined as follows:\n\n1. its objects are the objects of $\\mathcal{C}$;\n2. for objects $A$, $B$, its [[morphisms|859F0TFVA3]] are $\\Hom_{\\mathcal{C}^{\\mathrm{op}}}(A, B) = \\Hom_{\\mathcal{C}}(B, A)$; a morphism $f \\colon B \\to A$ of $\\mathcal{C}$ is written $f^{\\mathrm{op}} \\colon A \\to B$ when seen in $\\mathcal{C}^{\\mathrm{op}}$;\n3. the composite is $g^{\\mathrm{op}} \\circ f^{\\mathrm{op}} = (f \\circ g)^{\\mathrm{op}}$, and the identity of $A$ is $1_A^{\\mathrm{op}}$.\n\n## Remarks\n\nThe category axioms for $\\mathcal{C}^{\\mathrm{op}}$ follow from those of $\\mathcal{C}$ read backwards, and $(\\mathcal{C}^{\\mathrm{op}})^{\\mathrm{op}} = \\mathcal{C}$.\nExample: let $(P, \\le)$ be a [[preorder|0VABYWJWH9]], seen as a category with one morphism $a \\to b$ when $a \\le b$.\nIts opposite has one morphism $a \\to b$ when $b \\le a$, so it is the preorder $(P, \\ge)$.\nNon-example: the rule \"keep the arrows as they are and only reverse the order of composition\" does not give a category in general.\nFor $f \\colon A \\to B$ and $g \\colon B \\to C$, the expression $f \\circ g$ has no meaning in $\\mathcal{C}$ unless $C = A$.\nA morphism $f$ is a monomorphism in $\\mathcal{C}$ exactly when $f^{\\mathrm{op}}$ is an epimorphism in $\\mathcal{C}^{\\mathrm{op}}$.\n",
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