  For certain weighted locally convex spaces $X$ and $Y$ of one real variable
smooth functions, we characterize the smooth functions $\varphi: \mathbb{R} \to
\mathbb{R}$ for which the composition operator $C_\varphi: X \to Y, \, f
\mapsto f \circ \varphi$ is well-defined and continuous. This problem has been
recently considered for $X = Y$ being the space $\mathscr{S}$ of rapidly
decreasing smooth functions [1] and the space $\mathscr{O}_M$ of slowly
increasing smooth functions [2]. In particular, we recover both these results
as well as obtain a characterization for $X =Y$ being the space $\mathscr{O}_C$
of very slowly increasing smooth functions.
