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  "body": "\n## Summary\n\nA comonad on a category is the dual of a [[monad|FF6XXA6D1A]].\nIt is an endofunctor $G$ with a counit $\\varepsilon \\colon G \\Rightarrow \\mathrm{Id}$ and a comultiplication $\\delta \\colon G \\Rightarrow G G$ that satisfy coassociativity and counit laws.\nEvery adjunction $F \\dashv U$ gives a comonad $F U$ on the domain of $U$.\nComonads govern coalgebraic structure, descent theory and comonadic resolutions in homological algebra.\n\n## Explanations\n\n### L1 human\n\nA monad builds things up, while a comonad unfolds things.\nThe counit extracts a value, and the comultiplication duplicates the context around it.\nFor a fixed set $A$, the functor $X \\mapsto A \\times X$ on sets is a comonad.\nThe counit forgets the label $a$ in $(a, x)$, and the comultiplication copies the label, giving $(a, (a, x))$.\n",
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  "name": "Comonad",
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