Consistency models are promising generative models as they distill the multi-step sampling of score-based diffusion in a single forward pass of a neural network.Without access to sampling trajectories of a pre-trained diffusion model, consistency training relies on proxy trajectories built on an independent coupling between the noise and data distributions.Refining this coupling is a key area of improvement to make it more adapted to the task and reduce the resulting randomness in the training process.In this work, we introduce a novel coupling associating the input noisy data with their generated output from the consistency model itself, as a proxy to the inaccessible diffusion flow output.Our affordable approach exploits the inherent capacity of consistency models to compute the transport map in a single step.We provide intuition and empirical evidence of the relevance of our generator-induced coupling (GC), which brings consistency training closer to score distillation.Consequently, our method not only accelerates consistency training convergence by significant amounts but also enhances the resulting performance.