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This paper describes a biomimetic underwater fish robot prototype and its design methodology. The key question directing our design is the transfer of functionality from fish to a fish robot with respect to efficient mobility. We want to minimize mechanical complexity and achieve a low-cost fabrication. We argue for the case of morphological computation, i.e. achieving high mobility and efficiency by duplicating fish physical body structure. In this way, a possibly large part of the fish motion ability is outsourced to the embodiment, i.e. achieved by the interaction of the fish body parts and the water flow. This approach makes us focus on the material properties of a compliant tail propulsion mechanism. The tail is actuated by a single motor and we want to make it efficient by exploiting the energy propagation from the body to the surrounding fluid. We explain our design constraints, material choices and describe the design process. We draw conclusions about the relevance of our design parameters and design choices.