Copula and semicopula transforms

We characterize the transformation, defined for every copula $C$, by $C_h(x,y):=h^{(-1)}(C(h(x),h(y))$, where $x$ and $y$ belong to $[0,1]$ and $h$ is a strictly increasing and continuous function on $[0,1]$. We study this transformation also in the class of quasi-copulas and semicopulas