Characterize relatively internally projective dualizable modules beyond the ω1-compact case

Characterize the class of all relatively internally projective dualizable left modules over a rigid E1-monoidal category E within the category of E-linear dualizable categories Cat^E_dual, without assuming the ω1-compactness condition. In particular, identify necessary and sufficient conditions for a dualizable left E-module to be relatively internally projective beyond the proper and ω1-compact hypotheses of Theorem 3.6.

Background

The paper proves Theorem 3.6, establishing that if C is a dualizable left E-module that is proper and ω1-compact, then the functor Hom^E_dual(C, -) preserves short exact sequences; equivalently, such C is relatively internally projective over E in Cat^E_dual.

Section 3.3 draws an analogy with the Raynaud–Gruson criterion for projective modules and raises the issue of describing all relatively internally projective dualizable modules without the ω1-compactness restriction. The authors note that while properness plus ω1-compactness suffices, a general characterization beyond these hypotheses is not presently available in the paper.