natural transformation from infinite elementary linear group to infinity general linear group

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

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

is natural with respect to ring-homomorphisms

follows from:

• colim functor
  

• colim functor preserves naturality