Equality of C and C′ under coflatness and flatness on the same side

Determine whether, for a Hopf algebra H with bijective antipode, if H is left coflat over a left H-module factor coalgebra C and left flat over the associated right coideal subalgebra A = coCH, then C necessarily equals C′ = H/HA+; equivalently, prove or disprove that the canonical comparison C′ → C is an isomorphism under these hypotheses.

Background

Theorem 7.4 shows that when H is left coflat over C and right flat over A = coCH, H becomes left faithfully coflat over C, leading to C = H/HA+. However, when both coflatness over C and flatness over A hold on the same (left) side, the authors cannot conclude that C equals C′ = H/HA+.

Later, Section 10 revisits this via colocalization: under left coflatness over C and left flatness over A, the canonical surjection C′→C exhibits C′ as a perfect right colocalization of C, and the authors note no known examples where C′≠C, underscoring the open status of the equality question.

References

If H is left coflat over C and left flat over A = coCH, then H is left faithfully coflat over C' = H/HA+, but it is not clear whether C = C', in contrast to case (f) of Proposition 7.3.

— On Takeuchi's correspondence  (2501.06045 - Skryabin, 10 Jan 2025) in Section 7 (remark following Proposition 7.3 and before Theorem 7.4)

Takeuchi’s correspondence also gives a bijection between the set of (not necessarily normal) coideal subalgebras of $H$ satisfying a certain condition and the set of $H$-module factor coalgebras of $H$ satisfying a certain condition; see, {\it e.g.}, for an exposition. Can we interpret, formulate and verify the correspondence in the setting of tensor categories?

— A Galois connection between subalgebras and tensor subcategories  (2609.04073 - Shimizu et al., 3 Sep 2026) in Section 6, immediately before Section 6.1, Questions following Theorem CSA-Hopf-ideal-bijection