  This note explores the link between the q-model structure of flows and the
Ilias model structure of topologically enriched small categories. Both have
weak equivalences which induce equivalences of fundamental (semi)categories.
The Ilias model structure cannot be left-lifted along the left adjoint adding
identity maps. The minimal model structure on flows having as cofibrations the
left-lifting of the cofibrations of the Ilias model structure has a homotopy
category equal to the $3$-element totally ordered set. The q-model structure of
flows can be right-lifted to a q-model structure of topologically enriched
small categories which is minimal and such that the weak equivalences induce
equivalences of fundamental categories. The identity functor of topologically
enriched small categories is neither a left Quillen adjoint nor a right Quillen
adjoint between the q-model structure and the Ilias model structure.
