Extension of 2-coherence to reflective subuniverses

Determine whether Theorem 7.2, asserting 2-coherence of modality-induced left adjoints on coslices, extends to all reflective subuniverses.

Background

For a modality on a universe, the paper constructs an induced left adjoint on an arbitrary coslice and proves that it is 2-coherent. This yields colimits in the corresponding wild subcategory of modal types.

Reflective subuniverses generalize the modal subuniverses considered in the theorem. The paper explicitly asks whether the same 2-coherence result holds for every reflective subuniverse, rather than only those arising from the modality structure used in the proof.

References

There are a few open questions raised by our work. The simplest is the analysis of the dual statement that right adjoints preserve limits for wild categories, which should be similar to the one presented here. Another question is whether we can extend Theorem 7.2 to all reflective subuniverses.

— On Left Adjoints Preserving Colimits in Homotopy Type Theory  (2608.28473 - Hart, 28 Aug 2026) in Section 8, p. 20