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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category with [[zero morphisms|YF9F9EG6M3]], and let $f \\colon X \\to Y$ be a morphism of $\\mathcal{C}$.\nA *cokernel* of $f$ is an object $Q$ with a morphism $q \\colon Y \\to Q$ such that:\n\n1. $q \\circ f = 0$;\n2. for every morphism $u \\colon Y \\to T$ with $u \\circ f = 0$ there is a unique morphism $v \\colon Q \\to T$ with $u = v \\circ q$.\n\n## Remarks\n\nEquivalently, $q$ is a [[coequalizer|DVSTWGVCSZ]] of $f$ and $0 \\colon X \\to Y$; it is the [[kernel|S417YNEJ6M]] of $f$ in the opposite category.\nA cokernel is unique up to a unique isomorphism, is written $Y \\to \\operatorname{coker} f$, and is an [[epimorphism|RVXDSY04KG]].\n\nExample: in the category of [[modules|0NZ8JA25WK]] over a ring, the quotient map $q \\colon Y \\to Y / f(X)$ is a cokernel of $f$.\nIf $u \\circ f = 0$, then $u$ vanishes on $f(X)$ and so induces a unique $v \\colon Y / f(X) \\to T$ with $u = v \\circ q$.\nFor multiplication by $n \\ge 1$ on the [[abelian group|082GHNPJ53]] $\\Z$, the cokernel is the reduction $\\Z \\to \\Z / n\\Z$.\n\nNon-example: for multiplication by $2$ on $\\Z$, the map $\\Z \\to 0$ satisfies clause 1 but not clause 2.\nThe reduction $u \\colon \\Z \\to \\Z/2$ satisfies $u \\circ 2 = 0$ but does not factor through the zero group, since $u \\ne 0$.\n",
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