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  "body": "\n## Example\n\nThe *category of groups* $\\mathbf{Grp}$ has the [[groups|G5S1M9]] as objects and the [[group homomorphisms|0JYTG5K9Y4]] $f \\colon G \\to H$ as morphisms.\nComposition is the composition of maps, and the identity morphism of $G$ is the identity map $\\id_G$.\nThis is a [[category|A2DE66937D]].\nIf $f \\colon G \\to H$ and $g \\colon H \\to K$ are homomorphisms, then $g(f(xy)) = g(f(x)f(y)) = g(f(x))\\,g(f(y))$, so $g \\circ f$ is a homomorphism.\nThe identity map is a homomorphism, and composition of maps is associative, so both axioms of a category hold.\n\n## Remarks\n\nThe trivial group is a [[zero object|AQET9YHH6F]]: from each group there is exactly one homomorphism to it, and from it exactly one homomorphism to each group.\nThe [[product|S5744QZV4N]] of $G$ and $H$ is the [[direct product|00HF4RJ5QN]] $G \\times H$, and the [[coproduct|4FJ0S71P58]] is the [[free product|TM7QS7P7X9]] $G * H$.\nThe monomorphisms are the injective homomorphisms, and the epimorphisms are the surjective homomorphisms.\nForgetting the group structure gives a functor to the category of sets, whose left adjoint sends a set to the [[free group|020Z5SEN27]] on it.\n$\\mathbf{Grp}$ is not an [[abelian category|Y9CZ3MTNF1]]: for example, the coproduct $\\Z * \\Z$ differs from the product $\\Z \\times \\Z$, since the first group is not abelian.\n\n## Explanations\n\n### L1 human\n\nThe category of groups collects all groups and all structure-preserving maps between them into one object of study.\nMany group constructions become instances of general patterns there: the direct product is a categorical product and the free product is a coproduct.\nFor example, the map $\\Z \\to \\Z/2\\Z$ that sends an integer to its parity is a morphism of this category.\n",
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