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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category and let $\\mathcal{B}$ be a [[full subcategory|V3BFGKBJF7]] of $\\mathcal{C}$ with inclusion functor $I \\colon \\mathcal{B} \\to \\mathcal{C}$.\n$\\mathcal{B}$ is *reflective* in $\\mathcal{C}$ if $I$ has a [[left adjoint|TZHF6ZM92S]] $R \\colon \\mathcal{C} \\to \\mathcal{B}$.\nThe functor $R$ is called the *reflector*.\n\n## Remarks\n\nEquivalently, every object $c$ of $\\mathcal{C}$ has a morphism $\\eta_c \\colon c \\to R c$ with $R c$ in $\\mathcal{B}$.\nEvery morphism $c \\to b$ with $b$ in $\\mathcal{B}$ must factor through $\\eta_c$ in exactly one way.\nSome authors also ask $\\mathcal{B}$ to be closed under isomorphic objects.\nThe dual notion is a *coreflective* subcategory.\n\nExample: abelian groups form a reflective subcategory of groups, with reflector $R(H) = H/[H, H]$ and $\\eta_H$ the quotient map.\nEvery homomorphism $H \\to A$ with $A$ abelian kills all commutators $x y x^{-1} y^{-1}$, so it factors uniquely through $H/[H, H]$.\n\nNon-example: finite groups do not form a reflective subcategory of groups.\nA reflective subcategory closed under isomorphic objects contains every limit, formed in $\\mathcal{C}$, of a diagram in $\\mathcal{B}$.\nBut the product of countably many copies of $\\Z/2\\Z$ is an infinite group.\n",
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