An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and $2^{f}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer $n$, we study when the lightness of the induced map $2^{f}$ or $C_n(f)$ implies that $f$ is a homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.