  JWST is unveiling for the first time accreting black holes (BHs) with masses
of 10^6 - 10^7 Msun at z > 4, with the most distant residing in GNz11 at z =
10.6. Are we really surprised to find them in the nuclei of z = 5 - 11
galaxies? Here we predict the properties of 4 < z < 11 BHs and their host
galaxies considering an Eddington-limited (EL) and a super-Eddington (SE) BH
accretion scenario, using the Cosmic Archaeology Tool (CAT) semi-analytical
model. We calculate the transmitted spectral energy distribution of CAT
synthetic candidates, representative of the BH/galaxy properties of GNz11. We
also examine the possibility that the z = 8.7 galaxy CEERS-1019 could host an
active BH. We find that the luminosity of high-z JWST detected BHs are better
reproduced by the SE model, where BHs descend from efficiently growing light
and heavy seeds. Conversely, the host galaxy stellar masses are better matched
in the EL model, in which all the systems detectable with JWST surveys JADES
and CEERS descend from heavy BH seeds. We support the interpretation that the
central point source of GNz11 could be powered by a SE (lambda_Edd = 2 - 3)
accreting BH with mass 1.5 10^6 Msun, while the emission from CEERS-1019 is
dominated by the host galaxy; if it harbours an active BH, we find it to have a
mass of M_BH = 10^7 Msun, and to be accreting at sub-Eddington rates
(lambda_Edd = 0.5).
