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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $(X_i)_{i \\in I}$ be a family of objects of $\\mathcal{C}$ indexed by a set $I$.\nA *coproduct* of the family is an object $C$ with [[morphisms|859F0TFVA3]] $\\iota_i \\colon X_i \\to C$ for $i \\in I$, called *injections*, with the following property.\nFor every object $Y$ and every family of morphisms $f_i \\colon X_i \\to Y$, there is exactly one morphism $f \\colon C \\to Y$ with $f \\circ \\iota_i = f_i$ for all $i$.\nThe object $C$ is written $\\coprod_{i \\in I} X_i$, or $X_1 \\sqcup X_2$ for two objects, and $f$ is written $[f_1, f_2]$.\n\n## Remarks\n\nA coproduct is the [[colimit|BJS1CGTKQ5]] of the diagram whose shape is the set $I$ with only identity morphisms, and it is a [[product|S5744QZV4N]] in the opposite category.\nThe coproduct of the empty family is an initial object.\n\nExample: in the category of [[abelian groups|082GHNPJ53]], the [[direct sum|0S7W6EG32V]] $A \\oplus B$ with $\\iota_A(a) = (a, 0)$ and $\\iota_B(b) = (0, b)$ is a coproduct.\nGiven homomorphisms $f_A \\colon A \\to Y$ and $f_B \\colon B \\to Y$, the map $f(a, b) = f_A(a) + f_B(b)$ is a homomorphism because $Y$ is abelian, it satisfies $f \\circ \\iota_A = f_A$ and $f \\circ \\iota_B = f_B$, and it is forced since $(a, b) = (a, 0) + (0, b)$.\nIn the category of sets the coproduct is the disjoint union, and in the category of groups it is the [[free product|TM7QS7P7X9]].\n\nNon-example: in the category of fields, $\\mathbb{F}_2$ and $\\mathbb{F}_3$ have no coproduct, since no field receives homomorphisms from both: a field homomorphism preserves the characteristic.\n",
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