  Complex classical Cayley-Klein categories ${\bf A({\bj})}$, ${\bf B({\bj})}$,
${\bf C({\bj})}$ and ${\bf D({\bj})}$ are constructed by the method of
categorical extension of the complex classical Cayley-Klein groups
$SL(2n;{\bj};{\Bbb C}), SO(2n+1;{\bj};{\Bbb C}), Sp(2n;{\bj};{\Bbb C})$ and
$SO(2n;{\bj};{\Bbb C})$, respectively. The explicit construction of the
irreducible representations of the complex classical Cayley-Klein categories is
received. Completeness of lists of the irreducible representations of the
complex classical Cayley-Klein categories and the classification theorems are
proved.
