Inverse Morphism is Unique

Theorem
Let $\mathbf C$ be a metacategory.

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

Then $f$ admits a unique inverse morphism $g: D \to C$.

Proof
Since $f$ is an isomorphism, it admits at least one inverse morphism.

Now let $g, g': D \to C$ be two inverse morphisms for $f$.

Then:

In conclusion, $g = g'$.

Hence the result.