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We investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, by identifying conditions under which also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract determinization, these results enable the derivation of trace metrics starting from coalgebras in Set.doi:10.4230/lipics.calco.2015.35 dblp:conf/calco/BaldanBK015 fatcat:q737653gurd3lnw7go6u5uuzie