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Homenatural equivalence

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# natural equivalence

Let $F,G:\mathcal{C}\to\mathcal{D}$ be a pair of functors from the category $\mathcal{C}$ to the category $\mathcal{D}$. A natural transformation between functors $\tau:F\to G$ is called a natural equivalence (or a natural isomorphism) if there is a natural transformation $\sigma:G\to F$ such that $\tau\bullet\sigma={\rm id}_{G}$ and $\sigma\bullet\tau={\rm id}_{F}$ where ${\rm id}_{F}$ is the identity natural transformation on $F$, and composition $\bullet$ is the usual (vertical) composition on natural transformations.

Equivalently, one can define a natural equivalence from functors $F$ to $G$ to be a natural transformation $\tau$ such that for each object $A$ in $\mathcal{C}$, the morphism $\tau_{A}:F(A)\to G(A)$ is an isomorphism in $\mathcal{D}$.

## Mathematics Subject Classification

18-00*no label found*

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