Definition:Isomorphism of Categories

Definition
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 $\operatorname{id}_{\mathbf C}$
 * $F G: \mathbf D \to \mathbf D$ is the identity functor $\operatorname{id}_{\mathbf D}$

Also see

 * Equivalence of Categories