  We present a new zoom-in hydrodynamical simulation, "Erisbh", which follows
the cosmological evolution and feedback effects of a supermassive black hole at
the center of a Milky Way-type galaxy. ErisBH shares the same initial
conditions, resolution, recipes of gas cooling, star formation and feedback, as
the close Milky Way-analog "Eris", but it also includes prescriptions for the
formation, growth and feedback of supermassive black holes. We find that the
galaxy's central black hole grows mainly through mergers with other black holes
coming from infalling satellite galaxies. The growth by gas accretion is
minimal because very little gas reaches the sub-kiloparsec scales. The final
black hole is, at z=0, about 2.6 million solar masses and it sits closely to
the position of SgrA* on the MBH-MBulge and MBH-sigma planes, in a location
consistent with what observed for pseudobulges. Given the limited growth due to
gas accretion, we argue that the mass of the central black hole should be above
10^5 solar masses already at z~8. The effect of AGN feedback on the host galaxy
is limited to the very central few hundreds of parsecs. Despite being weak, AGN
feedback seems to be responsible for the limited growth of the central bulge
with respect to the original Eris, which results in a significantly flatter
rotation curve in the inner few kiloparsecs. Moreover, the disk of ErisBH is
more prone to instabilities, as its bulge is smaller and its disk larger then
Eris. As a result, the disk of ErisBH undergoes a stronger dynamical evolution
relative to Eris and around z=0.3 a weak bar grows into a strong bar of a few
disk scale lengths in size. The bar triggers a burst of star formation in the
inner few hundred parsecs, provides a modest amount of new fuel to the central
black hole, and causes the bulge of ErisBH to have, by z=0, a box/peanut
morphology.(Abridged)
