Smyth completion as bicompletion

We define a functor from topological quasi-uniform spaces and continuous, uniformly continuous maps to quasi-uniform spaces and uniformly continuous maps. This functor retains the same underlying set, and generalizes a construction due to Junnila (1978) and Künzi and Ferrario (1991) to obtain another quasi-uniformity, with respect to which the continuous uniformly continuous functions on the original structure are uniformly continuous; the functor then takes the maps to themselves. As a result, this functor takes the Smyth completion of a topological quasi-uniform space into the bicompletion of its image under the functor, and a topological quasi-uniform space is Smyth complete exactly when its image is bicomplete.