Every group is naturally isomorphic to its opposite group via the inverse map

Statement
Let $$G$$ be a group. Then, consider the opposite group of $$G$$, which is a group with the same underlying set, and such that the binary operation is defined by:

$$x * y := yx$$

In other words, products are taken with order reversed. Then, $$G$$ is isomorphic to the opposite group via the map $$g \mapsto g^{-1}$$.

This isomorphism is natural in the sense that it gives a natural isomorphism between the identity functor and the functor sending each group to its opposite group.

Related facts

 * Left and right coset spaces are naturally isomorphic

Facts used

 * 1) uses::Inverse map is involutive: This states that $$(xy)^{-1} = y^{-1}x^{-1}$$ for all $$x,y$$ in a group, and $$(x^{-1})^{-1} = x$$ for all $$x$$ in a group.