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# equivalence of categories

Let $C$ and $D$ be two categories with functors $F\colon C\to D$ and $G\colon D\to C$. The functors $F$ and $G$ are an equivalence of categories if there are natural isomorphisms $FG\cong\mathrm{id}_{D}$ and $GF\cong\mathrm{id}_{C}$.

Note, $F$ is left adjoint to $G$, and $G$ is right adjoint to $F$ as

$\hom_{D}(F(c),d)\stackrel{G}{\longrightarrow}\hom_{C}(GF(c),G(d))% \longleftrightarrow\hom_{C}(c,G(d)).$ |

And, $F$ is right adjoint to $G$, and $G$ is left adjoint to $F$ as

$\hom_{C}(G(d),c)\stackrel{F}{\longrightarrow}\hom_{D}(FG(d),F(c))% \longleftrightarrow\hom_{D}(d,F(c)).$ |

In practical terms, two categories are equivalent if there is a fully faithful functor $F\colon C\to D$, such that every object $d\in D$ is isomorphic to an object $F(c)$, for some $c\in C$.

Related:

EssentiallySurjective

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Definition

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Reference

## Mathematics Subject Classification

18A40*no label found*

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