  We derive the luminosity-temperature relation for the super-critically
accreting black holes (BHs) and compare it to the data on ultraluminous X-ray
sources (ULXs). At super-Eddington accretion rates, an outflow forms within the
spherization radius. We construct the accretion disc model accounting for the
advection and the outflow, and compute characteristic disc temperatures. The
bolometric luminosity exceeds the Eddington luminosity L_Edd by a logarithmic
factor 1+0.6 ln mdot (where mdot is the accretion rate in Eddington units) and
the wind kinetic luminosity is close to L_Edd. The apparent luminosity for the
face-on observer is 2-7 times higher because of geometrical beaming. Such an
observer has a direct view of the inner hot accretion disc, which has a peak
temperature T_max of a few keV in stellar-mass BHs. The emitted spectrum
extends as a power-law F_E ~ E**{-1} down to the temperature at the
spherization radius T_sp ~ mdot**(-1/2) keV. We associate T_max with a few keV
spectral components and T_sp with the soft, 0.1-0.2 keV components observed in
ULXs. An edge-on observer sees only the soft emission from the extended
envelope, with the photosphere radius exceeding the spherization radius by
orders of magnitude. The dependence of the photosphere temperature on
luminosity is consistent with that observed in the super-Eddington accreting
BHs SS 433 and V4641 Sgr. Strong outflows combined with the large intrinsic
X-ray luminosity of the central BH explain naturally the presence of the
photoionized nebulae around ULXs. An excellent agreement between the model and
the observational data strongly argues in favour of ULXs being super-critically
accreting, stellar-mass BHs similar to SS 433, but viewed close to the symmetry
axis.
