Note
Internal projectivity and properness of dualizable categories
Abstract
Relative internal projectivity implies properness for dualizable modules over a rigid -monoidal category. We prove this by constructing a strongly continuous localization from a proper module and transferring a criterion for properness through compact morphisms in the internal Hom. The converse fails over every nonzero rigid base. Over a field, an explicit compactly generated presheaf category gives a counterexample: a diagram of Laurent modules admits no lift, even up to retract, through the induced functor on dualizable internal Homs. We describe the resulting integer inequalities and show that every countable restriction of this diagram does lift.