Remark

Let A be a set endowed with a structure $\mathscr{S}$ of species $\Sigma$, and let R be an equivalence relation on A such that there exists on $E=A/R$ a quotient structure $\mathscr{S}'$ of $\mathscr{S}$ by R. Let $\varphi$ be the canonical mapping of A onto E. In general, there exists no section $s$ of $\varphi$ (Chapter II, § 3, no. 8) which is a morphism of E into A. Let us suppose that such a section $s$ exists, and moreover that there exists a structure $\mathscr{S}''$ induced by $\mathscr{S}$ on $s(E)$. Then, if $j$ denotes the canonical injection of $s(E)$ into A and if $s=j\circ f$, the bijection $f$ is an isomorphism of E onto $s(E)$. For $f$ is a morphism by the definition of induced structure, and $g=\varphi\circ j$ is a morphism of $s(E)$ onto E by reason of (MO$_{\mathrm{II}}$). Since $g\circ f$ and $f\circ g$ are the identity mappings of E and $s(E)$, respectively, the assertion is a consequence of CST8.