Dendrites and light mappings

It is shown that a metric continuum $X$ is a dendrite if and only if for every compact space (continuum) $Y$ and for every light confluent mapping $f: Y \to f(Y)$ such that $X \subset f(Y)$ there is a copy $X'$of $X$ in $Y$ for which the restriction $f\vert X': X' \to X$ is a homeomorphism. As a corollary it follows that only dendrites have the lifting property with respect to light confluent mappings. Other classes of mappings $f$ are also discussed. This is a continuation of a previous study by the authors (2000), where open mappings $f$ were considered.