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  "body": "\n## Definition\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]].\nThe functor $F$ is *essentially surjective* if, for every object $D$ of $\\mathcal{D}$, there is an object $A$ of $\\mathcal{C}$ with $F(A) \\cong D$.\nHere $\\cong$ means that there is an [[isomorphism|859F0TFVA3]] in $\\mathcal{D}$.\n\n## Remarks\n\nExample: let $K$ be a field and let $\\mathcal{C}$ be the [[full subcategory|V3BFGKBJF7]] of finite-dimensional [[vector spaces|000AYPA9AJ]] over $K$ spanned by the spaces $K^n$, $n \\ge 0$.\nIts inclusion into all finite-dimensional vector spaces over $K$ is essentially surjective, since a space of dimension $n$ is isomorphic to $K^n$.\nNon-example: the inclusion of the full subcategory of finite sets into the category of sets is not essentially surjective.\nAn infinite set is not isomorphic, that is, not in bijection with, any finite set.\nEvery functor that is surjective on objects is essentially surjective.\n",
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