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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be [[categories|A2DE66937D]].\nA *functor* $F \\colon \\mathcal{C} \\to \\mathcal{D}$ consists of:\n\n1. for each object $A$ of $\\mathcal{C}$, an object $F(A)$ of $\\mathcal{D}$;\n2. for each [[morphism|859F0TFVA3]] $f \\colon A \\to B$ of $\\mathcal{C}$, a morphism $F(f) \\colon F(A) \\to F(B)$ of $\\mathcal{D}$;\n\nsuch that:\n\n1. $F(1_A) = 1_{F(A)}$ for every object $A$ of $\\mathcal{C}$;\n2. $F(g \\circ f) = F(g) \\circ F(f)$ for all $f \\colon A \\to B$ and $g \\colon B \\to C$ in $\\mathcal{C}$.\n\n## Remarks\n\nSuch a functor is also called *covariant*.\nFunctors compose, and every category has an identity functor.\nA functor sends isomorphisms to isomorphisms: if $g \\circ f = 1_A$ and $f \\circ g = 1_B$, then $F(g) \\circ F(f) = 1_{F(A)}$ and $F(f) \\circ F(g) = 1_{F(B)}$.\nExample: the forgetful functor from [[groups|G5S1M9]] to sets sends a group to its underlying set and a [[group homomorphism|0JYTG5K9Y4]] to the same map.\nBoth conditions hold because identities and composites of homomorphisms are the identity and composite maps.\nNon-example: sending each group $G$ to its centre $Z(G)$ and each homomorphism to its restriction does not define a functor from groups to groups.\nLet $f \\colon \\Z/2 \\to S_3$ be the inclusion onto the subgroup generated by a transposition.\nIt maps the centre $\\Z/2$ outside the trivial centre of $S_3$.\nSo restriction gives no morphism $Z(\\Z/2) \\to Z(S_3)$, and the second item of the data fails.\n",
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