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Let $\mathbf C$ be a metacategory.
Let $f: X \to Y$ be a morphism of $\mathbf C$.
A morphism $g: Y \to X$ is said to be an inverse (morphism) for $f$ if and only if:
- $g \circ f = I_X$
- $f \circ g = I_Y$
where $I_X$ denotes the identity morphism on $X$.
It follows that $f$ is an isomorphism if and only if it has an inverse morphism.