Definition:Equivalence of Categories

Definition
Let $\mathbf C$ and $\mathbf D$ be metacategories.

An equivalence of $\mathbf C$ and $\mathbf D$ comprises:


 * Functors $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf C$


 * Natural isomorphisms $\alpha: G F \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf C}$ and $\beta: F G \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf D}$

Also defined as
Some sources call $F$ an equivalence if there exist $G, \alpha$ and $\beta$ as above.

Also see

 * Isomorphism of Categories