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  "body": "\n## Definition\n\nLet $K$ be a field and let $(C, \\Delta, \\varepsilon)$ be a [[coalgebra|A06RA8KB0T]] over $K$.\nA *right $C$-comodule* is a [[vector space|000AYPA9AJ]] $M$ over $K$ with a [[linear map|0NRYHJPQG2]] $\\rho \\colon M \\to M \\otimes C$, the coaction, such that:\n\n1. $(\\id_M \\otimes \\Delta) \\circ \\rho = (\\rho \\otimes \\id_C) \\circ \\rho$ as maps $M \\to M \\otimes C \\otimes C$.\n2. $(\\id_M \\otimes \\varepsilon) \\circ \\rho = \\id_M$, where $M \\otimes K$ is identified with $M$.\n\nA *left $C$-comodule* is defined in the same way with a map $M \\to C \\otimes M$.\n\n## Remarks\n\nExample: $C$ is a right comodule over itself with $\\rho = \\Delta$; clauses 1 and 2 are coassociativity and the counit law.\nExample: let $C = K[S]$ with $\\Delta(s) = s \\otimes s$ and $\\varepsilon(s) = 1$, and let $M = \\bigoplus_{s \\in S} M_s$ be a vector space graded by $S$.\nThen $\\rho(m) = m \\otimes s$ for $m \\in M_s$ is a coaction: both sides of clause 1 give $m \\otimes s \\otimes s$, and clause 2 gives $\\varepsilon(s) m = m$.\nEvery right $K[S]$-comodule arises in this way.\nNon-example: for $M \\ne 0$, the zero map $\\rho = 0$ satisfies clause 1 but fails clause 2.\nA morphism of comodules is a linear map $f \\colon M \\to N$ with $\\rho_N \\circ f = (f \\otimes \\id_C) \\circ \\rho_M$.\n",
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