Let $f : X \to Z$ be a continuous mapping
from a compact $C-$space $X$ onto a compact
space $Z$. Then there are a compact $C-$space $Y$ and
continuous mappings $g : X \to Y$ and $h : Y \to Z$
such that
$ {\dimC}Y \leq {\dimC}X, wY \leq wZ$ and $f = hg$.
Here ${\dimC}$ is Borst's transfinite extension of the covering
dimension $\dim$ in the class of compact $C$-spaces.