  In this paper the influence of chiral chemical potential $\mu_5$ on the
phenomenon of diquark condensation and phase structure of dense quark matter
altogether is contemplated in the framework of effective two-color and
two-flavor Nambu--Jona-Lasinio model. We show that the duality relations, found
$\mu_5=0$, are also valid for any value of $\mu_5\neq0$. Hence the dualities
seem to be a real fundamental feature of the phase diagram of two color quark
matter, and besides they are a very powerful tool of studying the phase
structure. In terms of dualities and the influence on the phase diagram chiral
imbalance $\mu_5$ stands alone from other chemical potentials. Two regimes in
the phase diagram could be discerned, the first one, rather small or moderate
values of chemical potentials $\mu_B$, $\mu_I$ and $\mu_{I5}$, where the
picture is rather concise and elegant: each of the above phenomena is in
one-to-one correspondence with some chemical potential ($\mu_{I5}$ induces
chiral symmetry breaking, $\mu_I$ propels charged pion condensation and $\mu_B$
leads to diquark condensation). In this regime chiral chemical potential
$\mu_5$ does not break this correspondence, it has a property of being
universal catalyzer, i.e. it {\it catalyzes/enhances} all the phenomena on
equal footing. The feature of chiral imbalance $\mu_5$ to catalyze chiral
symmetry breaking phenomenon was known before but its catalytic properties
appeared to be more universal and it can catalyze also diquark and charged pion
condensation. These happen due to the dual properties. In the second regime,
when several chemical potentials reach rather large values, one could observe a
rather complicated and rich phase structure, and $\mu_5$ can be a factor that
not so much catalyzes as {\it triggers} rather peculiar phases. For example, it
can lead to the formation of diquark condensation at zero baryon chemical
potential.
