  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.
