  AF Lep A+b is a remarkable planetary system hosting a gas-giant planet that
has the lowest dynamical mass among directly imaged exoplanets. We present an
in-depth analysis of the atmospheric composition of the star and planet to
probe the planet's formation pathway. Based on new high-resolution spectroscopy
of AF Lep A, we measure a uniform set of stellar parameters and elemental
abundances (e.g., [Fe/H] = $-0.27 \pm 0.31$ dex). The planet's dynamical mass
($2.8^{+0.6}_{-0.5}$ M$_{\rm Jup}$) and orbit are also refined using published
radial velocities, relative astrometry, and absolute astrometry. We use
petitRADTRANS to perform chemically-consistent atmospheric retrievals for AF
Lep b. The radiative-convective equilibrium temperature profiles are
incorporated as parameterized priors on the planet's thermal structure, leading
to a robust characterization for cloudy self-luminous atmospheres. This novel
approach is enabled by constraining the temperature-pressure profiles via the
temperature gradient $(d\ln{T}/d\ln{P})$, a departure from previous studies
that solely modeled the temperature. Through multiple retrievals performed on
different portions of the $0.9-4.2$ $\mu$m spectrophotometry, along with
different priors on the planet's mass and radius, we infer that AF Lep b likely
possesses a metal-enriched atmosphere ([Fe/H] $> 1.0$ dex). AF Lep b's
potential metal enrichment may be due to planetesimal accretion, giant impacts,
and/or core erosion. The first process coincides with the debris disk in the
system, which could be dynamically excited by AF Lep b and lead to planetesimal
bombardment. Our analysis also determines $T_{\rm eff} \approx 800$ K,
$\log{(g)} \approx 3.7$ dex, and the presence of silicate clouds and
dis-equilibrium chemistry in the atmosphere. Straddling the L/T transition, AF
Lep b is thus far the coldest exoplanet with suggested evidence of silicate
clouds.
