  Using Reedy techniques, this paper gives a correct proof of the left
properness of the q-model structure of flows. It fixes the preceding proof
which relies on an incorrect argument. The last section is devoted to fix some
arguments published in past papers coming from this incorrect argument. These
Reedy techniques also enable us to study the interactions between the path
space functor of flows with various notions of cofibrations. The proofs of this
paper are written to work with many convenient categories of topological spaces
like the ones of $k$-spaces and of weakly Hausdorff $k$-spaces and their
locally presentable analogues, the $\Delta$-generated spaces and the
$\Delta$-Hausdorff $\Delta$-generated spaces.
