\begin{definition}[Definition:Isomorphism of Categories]
Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be a functor.
Then $F$ is an '''isomorphism (of categories)''' {{iff}} there exists a functor $G: \mathbf C \to \mathbf D$ such that:
:$G F: \mathbf C \to \mathbf C$ is the identity functor $I_{\mathbf C}$
:$F G: \mathbf D \to \mathbf D$ is the identity functor $I_{\mathbf D}$
\end{definition}