  Let $L$ be a countable CW-complex and $F\colon X\to Y$ be upper
semicontinuous $UV^{[L]}$-valued mapping of a paracompact space $X$ to a
complete metric space $Y$. We prove that if $X$ is a C-space of extension
dimension $\ed X \le [L]$, then $F$ admits single-valued graph approximations.
For $L=S^n$ our result implies well-known approximation theorem for
$UV^{n-1}$-valued mappings of $n$-dimensional spaces. And for $L=\{\rm point\}$
our theorem implies a theorem of Ancel on approximations of $UV^\infty$-valued
mappings of C-spaces.
