The range of Archetype N was computed in Example FRAN to be \begin{equation*} \rng{T}=\spn{\set{ \colvector{1\\0\\0},\,\colvector{0\\1\\0},\,\colvector{0\\0\\1} } }\end{equation*}
Since the basis for this subspace is the set of standard unit vectors for $\complex{3}$ (Theorem SUVB), we have $\rng{T}=\complex{3}$ and by Theorem RSLT, $T$ is surjective.