  Donaldson-Thomas invariant is expressed as the weighted Euler characteristic
of the so-called Behrend (constructible) function. In \cite{Behrend} Behrend
introduced a DT-type invariant for a morphism. Motivated by this invariant, we
extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues.
We also discuss a categorification of the DT-type invariant for a morphism from
a bivariant-theoretic viewpoint, and we finally pose some related questions for
further investigations.
