= Solution
Lattice energy is transported by <phonons>; long-wavelength acoustic modes usually dominate low-temperature conduction. Their speed is set by elastic wave propagation. In the <Debye model> their volumetric heat capacity grows as $T^3$ at low temperature and approaches the classical lattice limit at high temperature. With boundary-limited mean free path, this gives a low-temperature $T^3$ conductivity; increasing phonon scattering can instead produce a peak and a declining high-temperature trend.
In metals, electrons near the <Fermi surface> carry both heat and charge. Their heat capacity is linear in T and their typical transport speed is the <Fermi velocity>, not a classical gas thermal speed. Degeneracy restricts the active thermal fraction but the large speed and long mean free path can make electronic heat conduction dominant. Insulators have few mobile electrons, so phonon conduction usually dominates. The approximate parallel-channel sum is $\kappa=\kappa_e+\kappa_{\rm ph}$; which channel dominates is material- and temperature-dependent. <Statistical Physics by David Tong>[Chapter3, phonon gas and low-temperature electron heat capacity]{ref}
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