Abstract
Given a copula $C$, we examine under which conditions on an order isomorphism $\psi$ of $[0,1]$, the distortion $C_{\psi}\colon [0,1]^2 o [0,1]$, $C_{\psi}(x,y)=\psi(C(\psi^{-1}(x)),\psi^{-1}(y))$, is again a copula. In particular, when the copula $C$ is totally positive of order $2$, we give a sufficient condition on $\psi$ which ensures that any distortion of $C$ by means of $\psi$ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behaviour in the tails.