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  "body": "\n## Summary\n\nAn equivalence of categories is a [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathcal{D}$ that has a functor $G \\colon \\mathcal{D} \\to \\mathcal{C}$ in the other direction.\nThe composites $G \\circ F$ and $F \\circ G$ must be naturally isomorphic to the identity functors.\nIt is the right notion of \"sameness\" for categories: it ignores how many copies of each object up to isomorphism a category has.\nA functor is an equivalence exactly when it is [[fully faithful|GRG8WEBSH5]] and [[essentially surjective|6ERBSBVF14]], by [[this theorem|WA60RW9E04]].\nEvery category is equivalent to its [[skeleton|A49QJYG669]].\n\n## Explanations\n\n### L1 human\n\nTwo categories are equivalent when they have the same objects up to isomorphism and the same arrows between them.\nIsomorphism of categories asks for more: an exact match of objects, which is rarely useful.\nFor example, the category of finite-dimensional real vector spaces has a huge number of objects.\nThe category with one object $n$ for each natural number, and with $m \\times n$ real matrices as the arrows from $n$ to $m$, has very few.\nYet the two are equivalent, which is why linear maps between finite-dimensional spaces can be studied as matrices.\n",
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