  The physics of baryons in halos, and their subsequent influence on the total
matter phase space, has a rich phenomenology and must be well understood in
order to pursue a vast set of questions in both cosmology and astrophysics. We
use the CAMELS simulation suite to quantify the impact of four different galaxy
formation parameters/processes (as well as two cosmological parameters) on the
concentration-mass relation, $c_{\rm vir} - M_{\rm vir}$. We construct a
simulation-informed nonlinear model for concentration as a function of halo
mass, redshift, and 6 cosmological/astrophysical parameters. This is done for
two galaxy formation models, IllustrisTNG and SIMBA, using 1000 simulations of
each. We extract the imprints of galaxy formation across a wide range in mass
$M_{\rm vir} \in [10^{11}, 10^{14.5}] M_{\rm \odot}/h$ and in redshift $z \in
[0,6]$ finding many strong mass- and redshift-dependent features. Comparisons
between the IllustrisTNG and SIMBA results show the astrophysical model choices
cause significant differences in the mass and redshift dependence of these
baryon imprints. Finally, we use existing observational measurements of $c_{\rm
vir} - M_{\rm vir}$ to provide rough limits on the four astrophysical
parameters. Our nonlinear model is made publicly available and can be used to
include CAMELS-based baryon imprints in any halo model-based analysis.
