  Fixpoints are ubiquitous in computer science as they play a central role in
providing a meaning to recursive and cyclic definitions. Bisimilarity,
behavioural metrics, termination probabilities for Markov chains and stochastic
games are defined in terms of least or greatest fixpoints. Here we show that
our recent work which proposes a technique for checking whether the fixpoint of
a function is the least (or the largest) admits a natural categorical
interpretation in terms of gs-monoidal categories. The technique is based on a
construction that maps a function to a suitable approximation and the
compositionality properties of this mapping are naturally interpreted as a
gs-monoidal functor. This guides the realisation of a tool, called UDEfix that
allows to build functions (and their approximations) like a circuit out of
basic building blocks and subsequently perform the fixpoints checks. We also
show that a slight generalisation of the theory allows one to treat a new
relevant case study: coalgebraic behavioural metrics based on Wasserstein
liftings.
