  Motivated by Smith's work \cite{Smith2003, Smith2016} on maps between
non-commu\-tative projective spaces of the form ${\rm Proj}_{nc} A$ in the
setting of non-commutative projective geometry developed by Rosenberg and Van
den Bergh, and the notion of schematicness introduced by Van Oystaeyen and
Willaert \cite{VanOystaeyenWillaert1995} to $\mathbb{N}$-graded rings with the
aim of formulating a non-commutative scheme theory \`a la Grothendieck
\cite{EGAII1961}, in this paper we consider a first approach to maps in the
Smith's sense in the more general setting of non-commutative projective spaces
over semi-graded rings defined by Lezama and Latorre \cite{LezamaLatorre2017}.
We extend Smith's key result \cite[Theorem 3.2]{Smith2003}, \cite[Theorem
1.2]{Smith2016} from the category of schematic $\mathbb{N}$-graded rings to the
category of schematic semi-graded rings.
