Necessity of full adjunctibility for integrals to live in Ω²C

Ascertain whether the weaker adjunctibility conditions in Proposition \ref{prop:dualizability} suffice to ensure that the universal integral and cointegral objects associated to the Hopf algebra H(X,f,g,α) live inside Ω²C (without passing to the Karoubi completion), or whether the stronger full adjunctibility conditions in Proposition \ref{prop:intcoint} are necessary.

Background

In the extra-dualizability analysis, the authors construct integrals and cointegrals for H(X,f,g,α) under full adjunctibility assumptions, paralleling classical results that relate Hopf algebras and the existence of universal (co)integrals. They note that, classically, such objects can require passage to the Karoubi completion.

The authors explicitly question whether weaker adjunctibility conditions already suffice to place the relevant integral objects inside Ω²C itself, without extra completion, or whether their stronger assumptions are essential. Answering this would refine the minimal hypotheses needed for integral/cointegral constructions in this (∞,3)-categorical framework.