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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category and let $F \\colon \\mathcal{C} \\to \\mathcal{C}$ be a [[functor|28NQ0KRHD3]].\nAn *$F$-coalgebra* is a pair $(A, \\alpha)$ of an object $A$ of $\\mathcal{C}$ and a morphism $\\alpha \\colon A \\to F(A)$.\nA *homomorphism* of $F$-coalgebras $f \\colon (A, \\alpha) \\to (B, \\beta)$ is a morphism $f \\colon A \\to B$ with $\\beta \\circ f = F(f) \\circ \\alpha$.\n\n## Remarks\n\nThe $F$-coalgebras and their homomorphisms form a category.\nAn $F$-coalgebra is the same as an algebra for the functor $F^{\\mathrm{op}}$ on $\\mathcal{C}^{\\mathrm{op}}$.\n\nExample: on the category of sets let $F(X) = \\N \\times X$, so $F(f) = \\id_{\\N} \\times f$.\nThe map $\\alpha \\colon \\N \\to \\N \\times \\N$ with $\\alpha(n) = (n, n + 1)$ makes $(\\N, \\alpha)$ an $F$-coalgebra.\nThe set $\\N^{\\N}$ of streams with $\\sigma(s) = (s_0, (s_1, s_2, \\dots))$ is the terminal $F$-coalgebra.\nThe unique homomorphism from $(\\N, \\alpha)$ sends $n$ to the stream $(n, n + 1, n + 2, \\dots)$.\n\nNon-example: the map $f \\colon \\N \\to \\N$ with $f(n) = n + 1$ is not a homomorphism from $(\\N, \\alpha)$ to itself.\nIndeed $\\alpha(f(n)) = (n + 1, n + 2)$, while $F(f)(\\alpha(n)) = (n, n + 2)$.\n",
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