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  "body": "\n## Statement\n\nLet $I$ be a small [[filtered category|975PMY4FG8]] and let $J$ be a finite category.\nLet $F \\colon I \\times J \\to \\mathbf{Set}$ be a functor on the [[product category|AFNE9C4NH6]] with values in the [[category of sets|WP0YB7VC59]].\nThen the canonical map\n$$\\operatorname{colim}_{i \\in I} \\lim_{j \\in J} F(i, j) \\longrightarrow \\lim_{j \\in J} \\operatorname{colim}_{i \\in I} F(i, j)$$\nis a bijection.\n\n## Remarks\n\nThe proof uses the explicit form of a filtered [[colimit|BJS1CGTKQ5]] of sets.\nIt is the disjoint union of the sets, where two elements are identified when they become equal at some later stage.\nA finite [[limit|CHS5AFJGWE]] involves only finitely many elements and finitely many equations.\nBecause $I$ is filtered, all of them can be moved to one common stage, which gives both injectivity and surjectivity.\n\nBoth hypotheses matter.\nFor $J$ infinite, take $I = \\N$, $J = \\N$ seen as a discrete category, and $F(n, j)$ a one-point set when $j \u003c n$ and empty otherwise.\nThen each $\\prod_j F(n, j)$ is empty, while each $\\operatorname{colim}_n F(n, j)$ is a point.\nSo the left side is empty and the right side is a point.\nFor $I$ not filtered, coproducts do not commute with products: $(A \\sqcup B) \\times (A \\sqcup B)$ is larger than $(A \\times A) \\sqcup (B \\times B)$ when $A$ and $B$ are not empty.\n\n## Explanations\n\n### L1 human\n\nA filtered colimit of sets is a kind of union of a growing family.\nA finite limit is a set of solutions of finitely many equations.\nThe theorem says that solving the equations and then taking the union gives the same as taking the union and then solving.\nThis works because any finite amount of data already lives at some stage of the growing family.\n\n### L2 human\n\nThe theorem explains why filtered colimits are so well behaved in algebra.\nFor example, a filtered colimit of groups or modules can be computed on underlying sets, because the group operations are maps out of finite products, which are finite limits.\nIt also shows that filtered colimits of exact sequences of modules are exact.\nThe same result holds in any locally finitely presentable category, and finitely presentable objects are defined by a closely related property of the hom functor.\n",
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