  A modal logic that is strong enough to fully characterize the behavior of a
system is called expressive. Recently, with the growing diversity of systems to
be reasoned about (probabilistic, cyber-physical, etc.), the focus shifted to
quantitative settings which resulted in a number of expressivity results for
quantitative logics and behavioral metrics. Each of these quantitative
expressivity results uses a tailor-made argument; distilling the essence of
these arguments is non-trivial, yet important to support the design of
expressive modal logics for new quantitative settings. In this paper, we
present the first categorical framework for deriving quantitative expressivity
results, based on the new notion of approximating family. A key ingredient is
the codensity lifting -- a uniform observation-centric construction of various
bisimilarity-like notions such as bisimulation metrics. We show that several
recent quantitative expressivity results (e.g. by K\"onig et al. and by
Fijalkow et al.) are accommodated in our framework; a new expressivity result
is derived, too, for what we call bisimulation uniformity.
