  We explain how the generalized Milnor-Wood inequality for reductive
representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie
group translates, under the non-abelian Hodge correspondence, into various
kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the
relation between the representation theoretic generalized Milnor-Wood
inequality and the various different versions of Milnor-Wood inequalities for
Higgs bundles that are known in the literature.
