Recently, two different copula-based approaches have been proposed to estimate the conditional quantile function of a variable \(Y\) with respect to a vector of covariates \(X\): the first estimator is related to quantile regression weighted by the conditional copula density, while the second estimator is based on the inverse of the conditional distribution function written in terms of margins and the copula. Using empirical processes, we show that even if the two estimators look quite different, they converge to the same limit. Also, we propose a bootstrap procedure for the limiting process in order to be able to construct uniform confidence bands around the conditional quantile function.