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  "body": "\n## Summary\n\nLet $F, G \\colon \\mathcal{C} \\to \\mathcal{D}$ be two [[functors|28NQ0KRHD3]].\nA natural transformation $\\alpha \\colon F \\Rightarrow G$ is a family of morphisms $\\alpha_A \\colon F(A) \\to G(A)$, one for each object $A$.\nThe family must commute with the images of all morphisms of $\\mathcal{C}$.\nIt makes precise the idea of a construction that is \"natural\", that is, defined in the same way for all objects without arbitrary choices.\nThe canonical map from a vector space to its double dual is natural.\nBy contrast, an isomorphism of a finite-dimensional space with its dual depends on a choice of basis.\nNatural transformations are the morphisms of [[functor categories|1X0WKFQX8K]] and enter the definitions of [[equivalence of categories|K7A04ZBD92]] and of [[adjoint functors|TZHF6ZM92S]].\n\n## Explanations\n\n### L1 human\n\nTwo functors give two pictures of the same category inside another category.\nA natural transformation is a way of moving from the first picture to the second, one object at a time.\n\"Natural\" means that it does not matter whether one first moves along an arrow and then switches pictures, or first switches and then moves.\nFor example, every vector space $V$ maps to its double dual by sending $v$ to \"evaluate at $v$\".\nThis rule uses no basis, and it is compatible with every linear map, so it is natural.\n",
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