In this note, we propose a novel adaptive fixed-time control scheme for output tracking problems of a class of multi-input multi-output (MIMO) nonlinear systems with asymmetric output constraint requirements, using a new universal barrier function. It is universal in the sense that the proposed scheme is a general one that also works for systems with symmetric constraints or without constraint requirements, without changing the control structure. Novel adaptive estimations and analysis are introduced to address system uncertainties in the fixed-time convergence settings. We show that under the proposed novel control scheme, each element in the system output tracking error vector of the MIMO nonlinear system can converge into small sets near zero with fixed-time convergence rate, while the asymmetric output constraint requirements on each element of the output tracking error are satisfied at all time. The proposed scheme can effectively deal with unmatched system uncertainties and uncertain gain functions. In the end, a simulation example on a two-degree-of-freedom robot manipulator is presented to demonstrate the efficacy of the proposed scheme.