Inverse of Algebraic Structure Isomorphism is Isomorphism/General Result

Theorem
Let $\phi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ and $\left({T, *_1, *_2, \ldots, *_n}\right)$ be algebraic structures.

Let $\phi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({T, *_1, *_2, \ldots, *_n}\right)$ be a mapping.

Then:
 * $\phi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({T, *_1, *_2, \ldots, *_n}\right)$ is an isomorphism

iff:
 * $\phi^{-1}: \left({T, *_1, *_2, \ldots, *_n}\right) \to \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ is also an isomorphism.