  We analyse the properties of the early-type dwarf galaxy population in the
Hydra I cluster. We investigate the galaxy luminosity function (LF), the
colour-magnitude relation (CMR), and the magnitude-surface brightness relation
down to M_V~-10 mag. Another goal of this study is to find candidates for
ultra-compact dwarf galaxies (UCDs). Two spectroscopic surveys performed with
Magellan I/LDSS2 and VLT/VIMOS, as well as deep VLT/FORS1 images in V and I
bands were examined. We identify cluster members by radial velocity
measurements and select other cluster galaxy candidates by their morphology.
One possible UCD candidate with M_V=-13.26 mag is found. Our sample of 100
morphologically selected dwarf galaxies defines a CMR that extends the CMR of
the giant cluster galaxies to the magnitude limit of our survey (M_V~-10 mag).
It matches the relations found for the Local Group and the Fornax cluster
dwarfs almost perfectly. The Hydra I dwarfs also follow a magnitude-surface
brightness relation similar to that of the LG dwarfs. Moreover, we observe a
continuous relation for dwarf galaxies and giant early-type galaxies when
plotting the central surface brightness \mu_0 of a S\'ersic model vs. the
galaxy magnitude. The effective radius is found to be largely independent of
the luminosity for M_V>-18 mag, being R_e~0.8 kpc. We derive a very flat
faint-end slope of the LF (\alpha = -1.13 \pm 0.04) from fitting a Schechter
function, whereas fitting a power law for M_V>-14 mag gives \alpha = -1.40 \pm
0.18. Our findings suggest that early-type dwarf and giant galaxies are the
same class of objects. The similarity of the dwarf galaxy scaling relations to
other environments implies that internal processes could be more important for
their global photometric properties than external influences. (abridged)
