natural transformation from elementary linear group to general linear group

Let \(R\) be a ring, \(E_n(R)\) the elementary linear group functor and \(\mathrm{GL}_n(R)\) the general linear group functor
Then the inclusion

\begin{align*} E_n(R) \rightarrow \mathrm{GL}_n(R) \end{align*}

is natural with respect to ring-homomorphisms

follows immediately from the definition, since for a ring homomorphism

\begin{align*} f: R \rightarrow S \end{align*}

the induced map

\begin{align*} E_n(f): E_n(R) \rightarrow E_n(S) \end{align*}

is just given by restricting the morphism

\begin{align*} \mathrm{GL}_n(f): \mathrm{GL}_n(R) \rightarrow \mathrm{GL}_n(S) \end{align*}