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  "body": "\n## Summary\n\nThe coproduct of a family of objects in a [[category|A2DE66937D]] is an object with an injection from each member of the family.\nAny family of morphisms out of the members factors uniquely through it.\nIt is the notion dual to the [[product|S5744QZV4N]], and it is the [[colimit|BJS1CGTKQ5]] of a diagram with no non-identity arrows.\nIt is the disjoint union for sets and topological spaces, the direct sum for abelian groups and modules, and the free product for groups.\n\n## Explanations\n\n### L1 human\n\nA map out of a coproduct is the same thing as a list of maps, one out of each piece.\nFor sets, the coproduct of $X$ and $Y$ is their disjoint union.\nA function on it is given by saying what it does on $X$ and what it does on $Y$, with no further constraint.\nFor abelian groups the answer is different: the coproduct of $A$ and $B$ is the [[direct sum|0S7W6EG32V]] $A \\oplus B$.\nThis works because homomorphisms out of $A$ and out of $B$ can be added together.\nIn a partially ordered set seen as a category, the coproduct of two elements is their least upper bound.\n",
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