Architecture of the Lyman-α forest clustering emulator. The blue arrow indicates the training direction, where the cNF optimizes a bijective mapping between the best-fitting parameters of the P_3D model to measurements from the TRAINING simulations and an eight-dimensional Normal distribution. The mapping is conditioned on cosmology and IGM physics, and performed using 12 consecutive affine coupling blocks. The green arrow denotes the emulation direction, where the cNF applies the inverse of the mapping to random samples from the base distribution to predict the value of the P_3D model parameters. Outside the cNF, FORESTFLOW introduces these parameters in Eq. (3) and (1) to obtain predictions for P_3D and P_1D, respectively.