  In this paper we consider the process of alignment of a spinning black hole
and a surrounding misaligned accretion disc. We use a simplified set of
equations, that describe the evolution of the system in the case where the
propagation of warping disturbances in the accretion disc occurs diffusively, a
situation likely to be common in the thin discs in Active Galactic Nuclei
(AGN). We also allow the direction of the hole spin to move under the action of
the disc torques. In such a way, the evolution of the hole-disc system is
computed self-consistently. We consider a number of different situations and we
explore the relevant parameter range, by varying the location of the warp
radius $R_{\rm w}$ and the propagation speed of the warp. We find that the
dissipation associated with the twisting of the disc results in a large
increase in the accretion rate through the disc, so that AGN accreting from a
misaligned disc are likely to be significantly more luminous than those
accreting from a flat disc. We compute explicitly the time-scales for the
warping of the disc and for the alignment process and compare our results with
earlier estimates based on simplified steady-state solutions. We also confirm
earlier predictions that, under appropriate circumstances, accretion can
proceed in a counter-aligned fashion, so that the accreted material will
spin-down the hole, rather than spinning it up. Our results have implication in
a number of different observational features of AGN such as the orientation and
shape of jets, the shape of X-ray iron lines, and the possibility of
obscuration and absorption of X-ray by the outer disc as well as the general
issue of the spin history of black holes during their growth.
