#naturality In [[Category theory|category theory]], natural equivalence is a concept that describes a relationship between two categories. It is a weaker notion than isomorphism, which requires a bijective correspondence between objects and morphisms. Formally, given two categories C and D, a natural equivalence between them is defined as a pair of functors F: C -> D and G: D -> C, along with two natural transformations η: 1C => GF and ε: FG => 1D, where 1C and 1D represent the identity functors on C and D respectively. The natural transformation η: 1C => GF is called the unit of the equivalence, while ε: FG => 1D is called the counit. These transformations satisfy certain coherence conditions known as triangle identities. Intuitively, a natural equivalence implies that C and D are "essentially the same" categories. This means that even though they may have different objects and morphisms, there exists a way to "translate" between them using the functors F and G. This translation should be compatible with composition of morphisms in both categories. Natural equivalences are useful in category theory because they allow us to compare different categories that may have different structures or properties. By establishing a natural equivalence between them, we can transfer knowledge or results from one category to another. It's important to note that not all categories are naturally equivalent. For example, if two categories have different numbers of objects or morphisms, they cannot be naturally equivalent. However, if there exists a natural equivalence between two categories, it implies that they share many important categorical properties. Overall, natural equivalence provides a powerful tool for understanding relationships between different categories by capturing their essential similarities while allowing for some differences in structure or composition. # References [[@NaturalEquivalence1945|General theory of natural equivalences]]