  This paper proves that the q-model structures of Moore flows and of
multipointed $d$-spaces are Quillen equivalent. The main step is the proof that
the counit and unit maps of the Quillen adjunction are isomorphisms on the
q-cofibrant objects (all objects are q-fibrant). As an application, we provide
a new proof of the fact that the categorization functor from multipointed
$d$-spaces to flows has a total left derived functor which induces a category
equivalence between the homotopy categories. The new proof sheds light on the
internal structure of the categorization functor which is neither a left
adjoint nor a right adjoint. It is even possible to write an inverse up to
homotopy of this functor using Moore flows.
