In category theory, equivalence of categories is a fundamental concept that captures the idea of two categories being "essentially the same" in a categorical sense. Two categories \( \mathcal{C} \) and \( \mathcal{D} \) are said to be equivalent if there exists a pair of functors between them that reflect a correspondence of their structural features, without necessarily being isomorphic.
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An equivalence consists of functors and with natural transformation isomorphisms and . A full and faithful functor which is essentially surjective, meaning every target object is isomorphic to an image object, is part of an equivalence under the usual choice convention. An isomorphism of categories is stronger: it has a strictly inverse functor.