Recent observations of supermassive black holes have brought us new information on their magnetospheres. In this study, we attempt a theoretical modelling of the coupling of black holes with their jets and discs, via three innovations. First, we propose a semi-analytical MHD description of a steady relativistic inflow–outflow structure characteristic to the extraction of the hole rotational energy. The mass-loading is ensured in a thin layer, the stagnation surface, by a two-photon pair production originating to a gamma-ray emission from the surrounding disc. The double flow is described near the polar axis by an axisymmetric meridionally self-similar MHD model. Secondly, the inflow and outflow solutions are crossing the MHD critical points and are matched at the stagnation surface. Knowledge of the MHD field on the horizon gives us the angular momentum and energy extracted from the black hole. Finally, we illustrate the model with three specific examples of double-flow solutions by varying the energetic interaction between the MHD field and the rotating black hole. When the isorotation frequency is half of the black hole one, the extracted Poynting flux is comparable to the one obtained using the force-free assumption. In two of the presented solutions, the Penrose process dominates at large colatitudes, while the third is Poynting flux dominated at mid-colatitudes. Mass injection rate estimations, from disc luminosity and inner radius, give an upper limit just above the values obtained for two solutions. This model is pertinent to describe the flows near the polar axis, where pair production is more efficient.