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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 *product* of the family is an object $P$ with [[morphisms|859F0TFVA3]] $\\pi_i \\colon P \\to X_i$ for $i \\in I$, called *projections*, with the following property.\nFor every object $Y$ and every family of morphisms $f_i \\colon Y \\to X_i$, there is exactly one morphism $f \\colon Y \\to P$ with $\\pi_i \\circ f = f_i$ for all $i$.\nThe object $P$ is written $\\prod_{i \\in I} X_i$, or $X_1 \\times X_2$ for two objects, and $f$ is written $(f_i)_{i \\in I}$ or $\\langle f_1, f_2 \\rangle$.\n\n## Remarks\n\nA product is the [[limit|CHS5AFJGWE]] of the diagram whose shape is the set $I$ seen as a category with only identity morphisms.\nSo it is unique up to unique isomorphism, and the product of the empty family is a terminal object.\n\nExample: in the category of sets, the [[Cartesian product|09CEDJYBKP]] $X_1 \\times X_2$ with the coordinate maps is a product.\nGiven $f_1 \\colon Y \\to X_1$ and $f_2 \\colon Y \\to X_2$, the map $f(y) = (f_1(y), f_2(y))$ satisfies $\\pi_i \\circ f = f_i$, and any such $f$ must have these coordinates.\nIn the category of groups the product is the [[direct product|00HF4RJ5QN]], and in the category of topological spaces it is the Cartesian product with the product topology.\n\nNon-example: in the category of fields, $\\mathbb{F}_2$ and $\\mathbb{F}_3$ have no product.\nField homomorphisms preserve the characteristic, so no field $P$ has morphisms to both $\\mathbb{F}_2$ and $\\mathbb{F}_3$.\n",
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