Definition:Equivalence of Categories

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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 $\beta: F G \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf D}$

Equivalent Categories

Let there exist an equivalence of categories between $\mathbf C$ and $\mathbf D$.

Then $\mathbf C$ and $\mathbf D$ are said to be equivalent, denoted $\mathbf C \simeq \mathbf D$.

Also defined as

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

Also see