We study a nonlinear lossless polarizer (NLP), a fiber-based device able to control the polarization of an optical signal while preserving its energy. The NLP exploits the lossless polarization attraction (LPA) generated by the Kerr interactions between the signal and a fully polarized continuous wave (CW) pump. By employing a copropagating pump, we show that the effectiveness of LPA depends on the joint action of the Kerr nonlinearity and the mutual delay between signal and pump. We find the optimal pump wavelength placement and demonstrate that true LPA occurs only within a limited range of delay values. Thus, we explain why the copropagating NLP is more flexible and power efficient compared with the traditional counterpropagating NLP.