Exercise 5, § 2

Let A, B be two sets endowed with structures $\mathscr{S}$, $\mathscr{S}'$ of the same species $\Sigma$. Let $f$ be a morphism of A into B, and let $g$ be a morphism of B into A. Let M (resp. N) be the set of all $x \in \mathrm{A}$ (resp. $y \in \mathrm{B}$) such that $g(f(x)) = x$ (resp. $f((g)y) = y$). Suppose that $\mathscr{S}$ (resp. $\mathscr{S}'$) induces on M (resp. N) a structure of species $\Sigma$. Show that M and N, endowed with these structures, are isomorphic.