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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *full subcategory* of $\\mathcal{C}$ is a [[subcategory|D8C0EYD9ED]] $\\mathcal{D}$ such that $\\Hom_{\\mathcal{D}}(A, B) = \\Hom_{\\mathcal{C}}(A, B)$ for all objects $A$, $B$ of $\\mathcal{D}$.\n\n## Remarks\n\nFor every subclass $S$ of $\\mathrm{Ob}(\\mathcal{C})$ there is exactly one full subcategory with object class $S$, the full subcategory *spanned* by $S$.\nExample: the [[abelian groups|082GHNPJ53]] form a full subcategory of the category of [[groups|G5S1M9]].\nA morphism between two abelian groups in the category of groups is a group homomorphism, and every such homomorphism is a morphism of the subcategory.\nNon-example: the subcategory of sets with all sets as objects and only the injective functions as morphisms is not full.\nThe constant map $\\set{0, 1} \\to \\set{0}$ is a morphism of the category of sets between two of its objects, but it is not injective.\n",
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