Cohomologically smooth morphisms for \'etale $\mathbb F_p$-sheaves in characteristic $p$

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