The steady Hiemenz stagnation-point flow of a flexible membrane moving bidirectionally, for a micropolar copper–water nanofluid model is studied. The flow field is a unison of wall transpiration, velocity slip and stagnation-point flow for a 2D case, generating unique or dual form of exact similarity solutions. The results reveal that a unique solution exists for the stretching sheet, depending entirely on a given set of parameters, while two solutions fall within the physically recognizable region for the case of a shrinking sheet. A set of new exact solutions characterized by the stretching/shrinking strength parameter d are also found with d considering different values. It is found that the numerical integration for skin friction, couple stress friction and temperature gradient leads to twofold solutions.