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  "body": "\n## Summary\n\nA [[category|A2DE66937D]] is well-powered when every object has only a set of [[subobjects|FS1FKAE89H]], not a proper class of them.\nThis size condition holds in most categories of everyday algebra and topology.\nIt is a hypothesis of the special adjoint functor theorem and of several existence results for factorisations and limits.\n\n## Explanations\n\n### L1 human\n\nIn a large category, there can be a proper class of monomorphisms into an object, because there is a proper class of sets that could serve as domains.\nAfter identifying monomorphisms that differ by an isomorphism, one usually gets back a set.\nFor example, the subobjects of a set $X$ correspond to the subsets of $X$, and these form the power set of $X$, which is a set.\nA well-powered category is one where this collapse to a set happens for every object.\n",
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