Definition:Isomorphism of Categories

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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) if and only if 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 $I_{\mathbf D}$

Isomorphic Categories

Let $F: \mathbf C \to \mathbf D$ be an isomorphism of categories.

Then $\mathbf C$ and $\mathbf D$ are said to be isomorphic, and we write $\mathbf C \cong \mathbf D$.

Also see

Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.