  Up-to techniques are a well-known method for enhancing coinductive proofs of
behavioural equivalences. We introduce up-to techniques for behavioural metrics
between systems modelled as coalgebras and we provide abstract results to prove
their oundness in a compositional way.
  In order to obtain a general framework, we need a systematic way to lift
functors: we show that the Wasserstein lifting of a functor, introduced in a
previous work, corresponds to a change of base in a fibrational sense. This
observation enables us to reuse existing results about soundness of up-to
techniques in a fibrational setting.
  We focus on the fibrations of predicates and relations valued in a quantale,
for which pseudo-metric spaces are an example. To illustrate our approach we
provide an example on distances between regular languages.
