  Aims. We aim to expand the atmospheric exploration of AF Lep b by modeling
all available observations obtained with SPHERE at VLT (between 0.95-1.65, at
2.105, and 2.253 $\mu$m, and NIRC2 at Keck (at 3.8 $\mu$m) with self-consistent
atmospheric models.
  Methods. To understand the physical properties of this exoplanet, we used
ForMoSA. This forward-modeling code compares observations with grids of
pre-computed synthetic atmospheric spectra using Bayesian inference methods. We
used Exo-REM, an atmospheric radiative-convective equilibrium model, including
the effects of non-equilibrium processes and clouds.
  Results. From the atmospheric modeling we derive solutions at a low effective
temperature of ~750 K. Our analysis also favors a metal-rich atmosphere (>0.4)
and solar to super-solar carbon-to-oxygen ratio (~0.6). We tested the
robustness of the estimated values for each parameter by cross-validating our
models using the leave-one-out strategy, where all points are used iteratively
as validation points. Our results indicate that the photometry point at 3.8
$\mu$m strongly drives the metal-rich and super-solar carbon-to-oxygen
solutions.
  Conclusions. Our atmospheric forward-modeling analysis strongly supports the
planetary nature of AF Lep b. Its spectral energy distribution is consistent
with that of a young, cold, early-T super-Jovian planet. We recover physically
consistent solutions for the surface gravity and radius, which allows us to
reconcile atmospheric forward modeling with evolutionary models, in agreement
with the previously published complementary analysis done by retrievals.
Finally, we identified that future data at longer wavelengths are mandatory
before concluding about the metal-rich nature of AF Lep b.
