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  "body": "\n## Example\n\nThe *category of rings* $\\mathbf{Ring}$ has the [[rings|0B1261B5ED]] with identity as objects and the [[ring homomorphisms|0VTH8KPCG2]] that send $1$ to $1$ as morphisms.\nComposition is the composition of maps, and the identity of $R$ is $\\id_R$.\nThis is a [[category|A2DE66937D]].\nA composite of two maps that preserve sums, products and $1$ again preserves them, and identity maps preserve them, so composites and identities are morphisms.\nComposition of maps is associative.\n\n## Remarks\n\nThe ring $\\Z$ is an [[initial object|EK85VSSRMR]]: for each ring $R$ the only homomorphism $\\Z \\to R$ sends $n$ to $n \\cdot 1_R$.\nThe zero ring is a terminal object, since every ring has exactly one map to it.\nThe zero ring is not initial, because a homomorphism from it must send $1 = 0$ to $1_R = 0_R$, which fails when $R$ is nonzero; so $\\mathbf{Ring}$ has no zero object.\nThe [[product|S5744QZV4N]] is the direct product $R \\times S$ with componentwise operations.\nThe commutative rings form a full subcategory $\\mathbf{CRing}$, in which the [[coproduct|4FJ0S71P58]] of $R$ and $S$ is the tensor product $R \\otimes_{\\Z} S$.\nThe inclusion $\\Z \\to \\Q$ is an epimorphism of $\\mathbf{Ring}$ that is not surjective: two homomorphisms out of $\\Q$ that agree on $\\Z$ agree on all fractions $m/n$.\n\n## Explanations\n\n### L1 human\n\nThe category of rings collects all rings and the maps between them that respect addition, multiplication and the unit.\nThe integers sit at the start of this category, since every ring receives exactly one map from $\\Z$.\nFor example, reduction modulo $5$ is a morphism $\\Z \\to \\Z/5\\Z$.\n",
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