Cohomologically smooth morphisms for étale ${\mathbb{F}}_p$-sheaves in characteristic $p$

We scrutinise the notions of cohomologically smooth morphisms and smooth objects for the six functor formalism of étale ${\mathbb{F}}_p$-sheaves on schemes in characteristic $p$. We show that only cohomologically étale morphisms are cohomologically smooth in this setting. This is complemented by a characterisation of cohomologically étale morphisms in arbitrary characteristic. In fact, we prove that such a morphism is already étale up to universal homeomorphism.