Does two variables have same correlation before and after applying copula?

I was reading a blog about copula. The post starts with two variables $X$, $Y$ sampling from a multivariate Gaussian with some kind of positive correlation, let's say $\rho$. Then applying copula (basically doing two times inverse cumulative function trick), we have new variables $X'=F^{-1}(\Phi(X))$ and $Y'=G^{-1}(\Phi(X))$, where $\Phi(x)$ is normal cumulative distribution function. While I understand $X'$ and $Y'$ then have marginal distribution $F(x)$ and $G(x)$, my question is does $corr(X', Y')$ equal $\rho$?

My guess is not necessarily, then does any method help correct variant correlation in case we want to maintain same correlation after applying copula?