Conjecture on preservation of left adjoint bicomodules and a ternary factorization system by Lan{p ∘ −}{p}

Prove that the functor Lan{p ∘ −}{p} sends left adjoint bicomodules to left adjoint bicomodules and preserves the ternary factorization system (Δ_ff, Δ_bo, Σ_dopf) on left adjoint bicomodules.

Background

The paper develops a P-enriched functorial selection-category construction S_C(p) ≅ Lan{p ∘ C}{p} and shows it respects bijective-on-objects and fully faithful morphisms, suggesting deeper structural preservation properties.

The authors conjecture that these observed properties are explained by a more fundamental preservation behavior of Lan{p ∘ −}{p} at the level of bicomodules and their factorization system, specifically the ternary factorization system (Δ_ff, Δ_bo, Σ_dopf). Establishing this would clarify the structural reasons behind the well-behavedness results proved for selection categories.