  Finite-temperature equation of state (EoS) and the composition of dense
nuclear and hypernuclear matter under conditions characteristic of neutron star
binary merger remnants and supernovas are discussed. We consider both neutrino
free-streaming and trapped regimes which are separated by a temperature of a
few MeV. The formalism is based on covariant density functional (CDF) theory
for the full baryon octet with density-dependent couplings, suitably adjusted
in the hypernuclear sector. The softening of the EoS with the introduction of
the hyperons is quantified under various conditions of lepton fractions and
temperatures. We find that $\Lambda$, $\Xi^-$, and $\Xi^0$ hyperons appear in
the given order with a sharp density increase at zero temperature at the
threshold being replaced by an extended increment over a wide density range at
high temperatures. The $\Lambda$ hyperon survives in the deep subnuclear
regime. The triplet of $\Sigma$s is suppressed in cold hypernuclear matter up
to around seven times the nuclear saturation density, but appears in
significant fractions at higher temperatures, $T\geq 20$ MeV, in both supernova
and merger remnant matter. We point out that a special isospin degeneracy point
exists where the baryon abundances within each of the three isospin multiplets
are equal to each other as a result of (approximate) isospin symmetry. At that
point, the charge chemical potential of the system vanishes. We find that under
the merger remnant conditions, the fractions of electron and $\mu$-on neutrinos
are close and are about 1\%, whereas in the supernova case, we only find a
significant fraction ($\sim$10\%) of electron neutrinos, given that in this
case, the $\mu$-on lepton number is zero.
