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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be a small category, let $\\mathcal{B}$ be a locally small category, and let $\\mathcal{C}$ be a category that has all small [[colimits|BJS1CGTKQ5]].\nLet $K \\colon \\mathcal{A} \\to \\mathcal{B}$ and $F \\colon \\mathcal{A} \\to \\mathcal{C}$ be functors.\nFor an object $b$ of $\\mathcal{B}$, let $(K \\downarrow b)$ be the [[comma category|PC6HG1JEGM]] of pairs $(a, f)$ with $a$ in $\\mathcal{A}$ and $f \\colon K(a) \\to b$.\nLet $P_b \\colon (K \\downarrow b) \\to \\mathcal{A}$ send $(a, f)$ to $a$.\nThen the [[left Kan extension|RN9D96ESWN]] of $F$ along $K$ exists, and for each object $b$ of $\\mathcal{B}$\n$$(\\mathrm{Lan}_K F)(b) = \\operatorname{colim} \\big( F \\circ P_b \\colon (K \\downarrow b) \\to \\mathcal{C} \\big).$$\n\n## Remarks\n\nThe comma category $(K \\downarrow b)$ is small because $\\mathcal{A}$ is small and $\\mathcal{B}$ is locally small, so the colimit exists.\nA morphism $g \\colon b \\to b'$ induces a functor $(K \\downarrow b) \\to (K \\downarrow b')$ by composing with $g$, and hence a map between the colimits; this makes $b \\mapsto (\\mathrm{Lan}_K F)(b)$ a functor.\nThe component $\\eta_a$ is the colimit injection at the object $(a, \\id_{K(a)})$.\nThe universal property is checked by turning a natural transformation $F \\Rightarrow G \\circ K$ into a compatible family of cocones, one for each $b$.\n\nDually, suppose that $\\mathcal{C}$ has all small [[limits|CHS5AFJGWE]].\nThen $(\\mathrm{Ran}_K F)(b)$ is the limit of $F$ over the comma category $(b \\downarrow K)$ of pairs $(a, f \\colon b \\to K(a))$.\nWhen $K$ is fully faithful, the unit $\\eta \\colon F \\Rightarrow (\\mathrm{Lan}_K F) \\circ K$ is an isomorphism.\nThe reason is that $(a, \\id_{K(a)})$ is then a terminal object of $(K \\downarrow K(a))$.\n\nExample: view a group $G$ and a subgroup $H$ as one-object categories, and let $K$ be the inclusion.\nLet $F$ be a representation of $H$ on a vector space $V$ over a field $k$.\nThen the formula gives the induced representation $k[G] \\otimes_{k[H]} V$ of $G$.\n\n## Explanations\n\n### L1 human\n\nThe formula builds the extended functor one object at a time.\nAt an object $b$, it collects all values $F(a)$ for which $K(a)$ maps into $b$ and glues them together by a colimit.\nSo the extension at $b$ is the best approximation of $F$ seen from everything that lies over $b$.\n\n### L2 human\n\nThis is the standard way to show that Kan extensions exist, and it is the version used in most applications.\nTaking $\\mathcal{B} = \\mathbf{1}$ recovers the colimit of $F$.\nTaking $K$ to be the Yoneda embedding of $\\mathcal{A}$ extends $F$ to a colimit-preserving functor on presheaves.\nThis is the basis of the nerve and realization adjunctions.\nA Kan extension computed by this formula is called pointwise.\nIn the language of coends the formula reads $(\\mathrm{Lan}_K F)(b) = \\int^{a} \\mathcal{B}(K(a), b) \\cdot F(a)$.\nHere $S \\cdot c$ is the coproduct of copies of $c$ indexed by the set $S$.\n",
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