\begin{definition}[Definition:Equivalence of Categories]
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 $\beta: F G \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf D}$
\end{definition}