  Behavioural distances of transition systems modelled via coalgebras for
endofunctors generalize traditional notions of behavioural equivalence to a
quantitative setting, in which states are equipped with a measure of how
(dis)similar they are. Endowing transition systems with such distances
essentially relies on the ability to lift functors describing the one-step
behavior of the transition systems to the category of pseudometric spaces. We
consider the category theoretic generalization of the Kantorovich lifting from
transportation theory to the case of lifting functors to quantale-valued
relations, which subsumes equivalences, preorders and (directed) metrics. We
use tools from fibred category theory, which allow one to see the Kantorovich
lifting as arising from an appropriate fibred adjunction. Our main
contributions are compositionality results for the Kantorovich lifting, where
we show that that the lifting of a composed functor coincides with the
composition of the liftings. In addition, we describe how to lift distributive
laws in the case where one of the two functors is polynomial (with finite
coproducts). These results are essential ingredients for adapting
up-to-techniques to the case of quantale-valued behavioural distances. Up-to
techniques are a well-known coinductive technique for efficiently showing lower
bounds for behavioural distances. We illustrate the results of our paper in two
case studies.
