  The behaviour of statistical relational representations across differently
sized domains has become a focal area of research from both a modelling and a
complexity viewpoint.Recently, projectivity of a family of distributions
emerged as a key property, ensuring that marginal probabilities are independent
of the domain size. However, the formalisation used currently assumes that the
domain is characterised only by its size. This contribution extends the notion
of projectivity from families of distributions indexed by domain size to
functors taking extensional data from a database. This makes projectivity
available for the large range of applications taking structured input. We
transfer key known results on projective families of distributions to the new
setting. This includes a characterisation of projective fragments in different
statistical relational formalisms as well as a general representation theorem
for projective families of distributions. Furthermore, we prove a
correspondence between projectivity and distributions on countably infinite
domains, which we use to unify and generalise earlier work on statistical
relational representations in infinite domains. Finally, we use the extended
notion of projectivity to define a further strengthening, which we call
$\sigma$-projectivity, and which allows the use of the same representation in
different modes while retaining projectivity.
