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  "body": "\n## Definition\n\nLet $K \\colon \\mathcal{A} \\to \\mathcal{B}$ and $F \\colon \\mathcal{A} \\to \\mathcal{C}$ be [[functors|28NQ0KRHD3]].\nA *left Kan extension* of $F$ along $K$ is a functor $L \\colon \\mathcal{B} \\to \\mathcal{C}$ together with a [[natural transformation|KYCPCC57FT]] $\\eta \\colon F \\Rightarrow L \\circ K$ with this property.\nFor every functor $G \\colon \\mathcal{B} \\to \\mathcal{C}$ and every natural transformation $\\alpha \\colon F \\Rightarrow G \\circ K$, there is a unique natural transformation $\\beta \\colon L \\Rightarrow G$ such that $\\alpha = (\\beta K) \\circ \\eta$.\nHere $\\beta K$ is the natural transformation with components $\\beta_{K(a)}$ for $a$ in $\\mathcal{A}$.\n\nA *right Kan extension* of $F$ along $K$ is a functor $R \\colon \\mathcal{B} \\to \\mathcal{C}$ together with a natural transformation $\\varepsilon \\colon R \\circ K \\Rightarrow F$ with this property.\nFor every functor $G \\colon \\mathcal{B} \\to \\mathcal{C}$ and every natural transformation $\\alpha \\colon G \\circ K \\Rightarrow F$, there is a unique natural transformation $\\beta \\colon G \\Rightarrow R$ such that $\\alpha = \\varepsilon \\circ (\\beta K)$.\n\n## Remarks\n\nThe usual notation is $\\mathrm{Lan}_K F$ for the left and $\\mathrm{Ran}_K F$ for the right Kan extension.\nBoth are unique up to a unique natural isomorphism when they exist.\nSuppose these extensions exist for all $F$.\nThen, between the [[functor categories|1X0WKFQX8K]], $\\mathrm{Lan}_K$ is left adjoint and $\\mathrm{Ran}_K$ is right adjoint to the restriction $G \\mapsto G \\circ K$ from $[\\mathcal{B}, \\mathcal{C}]$ to $[\\mathcal{A}, \\mathcal{C}]$.\n\nExample: let $\\mathcal{B} = \\mathbf{1}$ be the category with one object and one morphism, and $K$ the unique functor.\nA functor $G \\colon \\mathbf{1} \\to \\mathcal{C}$ is an object $c$, and $G \\circ K$ is the constant functor at $c$.\nA natural transformation $F \\Rightarrow G \\circ K$ is then a cocone from $F$ to $c$.\nSo the clause of the definition says exactly that $\\eta$ is a universal cocone, and $\\mathrm{Lan}_K F$ is the colimit of $F$.\n\nNon-example: let $\\mathcal{A}$ be the empty category, $\\mathcal{B} = \\mathbf{1}$ and $\\mathcal{C}$ the category of sets.\nThen $F$ and $\\eta$ are empty, and a left Kan extension is an initial object of $\\mathcal{C}$, namely the empty set.\nThe constant functor $L$ at a one-point set is not a left Kan extension.\nFor $G$ the constant functor at the empty set there is no natural transformation $L \\Rightarrow G$, so the existence part of the clause fails.\n",
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