  With its last observing run, the LIGO, Virgo, and KAGRA collaboration has
detected almost one hundred gravitational waves from compact binary
coalescences. A common approach to studying the population properties of the
observed binaries is to use phenomenological models to describe the spin, mass,
and redshift distributions. More recently, with the aim of providing a clearer
link to astrophysical processes forming the observed compact binaries
coalescences, several authors have proposed to employ synthetic catalogs for
population studies. In this paper, we review how to employ and interpret
synthetic binary catalogs for gravitational-wave progenitors studies. We
describe how to build multi-channel merger rates and describe their associated
probabilities focusing on stellar progenitor properties. We introduce a method
to quantify the match between the phenomenological reconstruction of merger
rates with synthetic catalogs. We detail the implementation of synthetic
catalogs for multi-channel hierarchical Bayesian inference, highlighting
computational aspects and issues related to hyper-prior choice. We find that
when inferring stellar progenitors' properties from gravitational-wave
observations, the relative efficiency in compact objects production should be
taken into account. Finally, by simulating binary black hole detections with
LIGO and Virgo sensitivity expected for the O4 observing run, we present two
case studies related to the inference of the common envelope efficiency and
progenitor metallicity of the binary black holes. We finally discuss how
progenitors' properties can be linked to binary black hole properties.
