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In our paper we give a characterization of (set-valued) maps which are minimal usco and minimal cusco simultaneously. Let be a topological space and be a Banach space. We show that there is a bijection between the space of minimal usco maps from to and the space of minimal cusco maps from to , and we study this bijection with respect to various topologies on underlying spaces. Let be a Baire space and be a Banach space. Then and are homeomorphic, where is the topology of uniform convergence.