Cohomologically smooth morphisms for \'etale Fₚ-sheaves in characteristic p

We scrutinise the notions of cohomologically smooth morphisms and smooth objects for the six functor formalism of \'etale Fₚ-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.