It is quite clear in many cases how to construct random vectors having specified copulas, e.g. the Gaussian copula, for example starting from a multivariate normal random vector (obtained for example with the Choleski factorization, etc.), and then producing a vector of standard uniforms $(U_1, \ldots, U_n)$ having cumulative distribution function equal to the Gaussian copula $C^{Ga}_{\Sigma}$.
My question is: how to do the converse? That is, suppose that I have a copula (for example the Gaussian copula above), $C^{Ga}_{\Sigma}$. I want to generate a realization of this copula, i.e., a random vector $U=(U_1, \ldots, U_n)$ having $C^{Ga}_{\Sigma}$ as copula. I haven't found any algorithm which accomplishes this. Do you know any such algorithm?