Extriangulated nature of duality functors in rigid tensor extriangulated categories

Ascertain whether, in a rigid tensor extriangulated category (i.e., a tensor extriangulated category in which every object is strongly dualisable), the duality functors given by left and right duals are extriangulated functors, meaning that they preserve extriangles and admit the requisite natural transformations on extension bifunctors.

Background

In a closed symmetric monoidal category, strongly dualisable objects have two-sided duals, and rigidity means every object is strongly dualisable. The paper establishes consequences of strong dualisability for extension groups and adjunctions in extriangulated settings and proves that, when every object is strongly dualisable, projective and injective objects coincide.

However, despite these structural results, it remains unresolved whether the duality functors arising from left or right duals themselves form extriangulated functors—that is, whether they are compatible with extriangles and the associated extension structures. This question is straightforward in abelian settings via kernels/cokernels but is unclear in the general extriangulated context.